EXISTENCE AND NONEXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM


 Patience Thomas
 1 years ago
 Views:
Transcription
1 EXISTENCE AND NONEXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM SALOMÓN ALARCÓN, JORGE GARCÍAMELIÁN AND ALEXANDER QUAAS Abtract. In thi paper we conider the nonlinear elliptic problem u + αu = g( u ) + λh(x) in Ω u = on Ω, where Ω i a mooth bounded domain of R N, α, g i an arbitrary C increaing function and h C (Ω) i nonnegative. We completely analyze the exitence and nonexitence of (poitive) claical olution in term of the parameter λ. We how that there exit olution for every λ when α = and the integral =, or α > and the integral g() =. Converely, when the repectively integral converge g() and h i nontrivial on Ω, exitence depen on the ize of λ. Moreover, nonexitence hol for large λ. Our proof mainly rely on comparion argument, and on the contruction of uitable uperolution in annuli. Our reult include ome cae where the function g i uperquadratic and till exitence hol without auming any mallne condition on λ.. Introduction The concern of the preent paper i the exitence and nonexitence of olution to the following nonlinear elliptic problem: u + αu = g( u ) + λh(x) in Ω (.) u = on Ω, where Ω i a bounded domain of cla C 2,η of R N for ome η (, ), α and g C (R) i increaing with g() =. The function h C (Ω), will be nonnegative, while λ will be regarded a a poitive parameter. We will focu our attention on general function g, obtaining harp condition which imply: either (a) problem (.) ha a unique olution for every λ or (b) there exit a critical ize of λ that divide exitence from nonexitence for (.) when h in Ω. Thi type of problem ha been extenively tudied. Here we give a quick review on the topic, other reference can be found in the paper quoted below. The pioneering work on the ubject eem to be due to Serrin [29], Amann and Crandall [6] and Lion [24]. The cae α > i conidered in [2] and [3], where exitence hol when g ha at mot quadratic growth, ee alo [4]. The cae α = and g(t) = t 2 wa tudied for example in [7] and [8] (ee alo [2] and [2]). For related reult ee [] and []. More recently alo related fully nonlinear equation are conidered in [3] (ee alo the cae α < in [22], where multiplicity reult are obtained).
2 2 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS We finally mention that a tarting point of our work can be found in [3]; actually we olve a problem given in that paper, ee Remark on page 29 there. More precie information on our contribution with repect to the known reult are given in the remark after our main theorem. By a olution to (.) we mean a function u C 2 (Ω) C (Ω) verifying the equation in the claical ene. Remark that, on one hand, tandard boottrapping give u C 2,η (Ω), while on the other hand olution are trictly poitive in Ω by the maximum principle, ince u + αu in Ω. An important remark with regard to problem (.) i that uniquene of olution hol by the comparion principle (cf. for intance Theorem. in [9] or the reult in [27] and [6]). Thu we only need to how exitence and nonexitence of olution. For other uniquene reult ee for example [8], [9] and []. Oberve alo that nonuniquene hol with le regularity on the olution, ee for intance [2]. Let u tate now our main reult. We begin with the cae α =. It turn out that the exitence of olution depen on the condition (.2) More preciely: g() =. Theorem. Aume g C (R) i increaing with g() =, while h C (Ω) i uch that h in Ω. If α = then: (i) If (.2) hol, there exit a unique olution to (.) for every λ >. (ii) If (.2) doe not hold and h on Ω, then there exit Λ > uch that for λ (, Λ) problem (.) ha a unique olution, while there are no olution when λ > Λ. Remark. (a) The nonexitence part in (ii) i already proved in the cae where g i convex (ee Theorem 2. in [3]). In the particular cae g(t) = t p with p > 2, ee [2] for exitence when h i a meaure and λ i mall. (b) Part (ii) of Theorem exten the cae g(t) = t 2 of the above quoted paper. We now turn to the cae α >. exitence of olution i (.3) In thi cae, the condition for the =. g() Oberve that condition (.3) i implied by (.2). We have: Theorem 2. Aume g C (R) i increaing with g() =, while h C (Ω) i uch that h in Ω. If α >, then: (i) If (.3) hol, there exit a unique olution to (.) for every λ >. (ii) If (.3) doe not hold and h in Ω, then there exit Λ > uch that for λ (, Λ) problem (.) admit a unique olution, while no olution exit when λ > Λ.
3 GENERAL CONVECTION TERM 3 Remark 2. (a) The nonexitence part in (ii) i already proved in the particular cae g(t) = t p with p > 2, ee Propoition 2.3 in [3]. (b) Part (i) in Theorem 2 applie for intance to g(t) = t 2 ln( + t), which i uperquadratic, and o exitence hol without the mallne retriction on the right hand ide. Thi mean that in the etting of claical olution, and with mooth data, mot of the previou exitence theorem are not optimal with repect to the growth condition in g, ince at mot quadratic behavior i required in the cae α >. (c) Part (i) in our two theorem anwer an open quetion tated in [3] (ee Remark in page 29 of that paper). We are indeed a little bit more precie here, ince our optimal condition are different for the cae α = and α >. When g(t) = t part (i) hol in both theorem, thi particular cae already being covered in [3] (ee Theorem 3. there). (d) The cae g(t) = O(t) ha been uually the reference cae for a general olvability reult (Part (i) in Theorem or Theorem 2); ee for example [5], and [5] (note that the ymmetrization approach reduce the problem to a radial one, which i related to our approach). The uperlinear model cae g(t) = t q, q > i deeply tudied in [2], in particular a far a neceary condition for the exitence are concerned. In [26], the cae g(t) = t q with the limit α i decribed and the maximal contant Λ i characterized in term of tochatic tate contraint ergodic problem. For the uperquadratic cae, ee alo [7], where the exitence of a generalized vicoity olution i proved when λ >, though thi olution i not claical and in particular may not attain the boundary datum. Let u mention in paing that the poitivity condition on h i only impoed in order to implify the preentation. In particular, it i relevant for the nonexitence reult only. For a function h which take both ign we till may aert the exitence of olution for every λ > in cae (i) in both theorem and for mall λ in cae (ii), although in thi lat ituation the reult are not expected to be optimal. Notice alo that when h i negative, the change of u by u in (.) amount to replacing g by g. In mot of the previou work, no ditinction i made between the two cae, but the reult are far from optimal. Here we have decided to retrict our attention to nonnegative h (hence poitive g) for definitene. On the other hand, we believe that the proof can be adapted to deal with ome more general operator than the Laplacian, for intance the p Laplacian or even ome fully nonlinear operator which depend on the econd derivative of the olution. The baic idea to prove exitence of olution to (.) come from [24] (ee alo [25] and [23]). It conit in truncating the term g( u ) in order to obtain a problem in a claical etting, i. e. with ubquadratic growth in the gradient. Then the tandard method of ub and uperolution can be ued to get a olution to the truncated problem, and the final tep i to how that the olution to the truncated problem i indeed a olution to the original one. Thi can be achieved by obtaining appropriate etimate for the gradient u of the olution u in Ω. Thee etimate are a conequence
4 4 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS of a kind of maximum principle for u 2 + u 2, o that everything i reduced to etimating u on Ω. Thi can be done by comparing with a uitable uperolution. It i important to notice that our approach doe not rely in obtaining a uperolution ū to (.) which vanihe on the whole Ω, omething which i required to apply Theorem III. in [24]. Rather, we contruct the uperolution by analyzing problem (.) in an annulu which after a uitable tranlation i tangent to Ω at every fixed x Ω. Thi enable u to deal with a radial problem which i in ome ene integrable, o we are able to find condition which are both neceary and ufficient for exitence. The ret of the paper i organized a follow: in Section 2 we contruct uperolution to (.) in the particular cae where Ω i an annulu. Section 3 i dedicated to how nonexitence of olution to (.) when Ω i a ball. Finally, in Section 4 we deal with the proof of Theorem and Superolution for problem in annuli It will be proved in Section 4 that the exitence of a radial uperolution to problem (.) poed in an annulu when h i contant uffice to enure the exitence of a olution to (.). Thu thi ection will be dedicated to contruct a poitive radial function u verifying (2.) (r N u ) r N ( αu + g( u ) + c), R < r < R 2 u(r ) =, u(r 2 ), for uitable value of c, depending on whether α = or α > and alo on the integrability condition on g at infinity conidered in the Introduction. In what follow, R 2 > R > will be fixed. Lemma 3. Aume g C (R) i increaing with g() = and α =. Then, if (2.2) g() =, for every c > there exit a poitive radial function u verifying (2.). If (2.2) doe not hold, the exitence of uch a function alo follow provided c i mall enough. Proof. Introducing the change of variable log r N = 2 (2.3) =, N 3, N 2 rn 2 and denoting u(r) = v(), (2.) i tranformed into v r 2(N ) ( g ( ( v ) ) + c ) r N v(a) =, v(b),
5 GENERAL CONVECTION TERM 5 where a = log R, b = log R 2 when N = 2, while a = N 2 R N 2 2 N 2 R N 2, b = if N 3. Since g i increaing and poitive, it uffice to have ( ( v R 2(N ) 2 g v(a) =, v(b). R N ) ) v + c Setting w() = v( + a), thi ugget to conider the onedimenional autonomou initial value problem ( ( ) ) w = R 2(N ) 2 g w + c (2.4) R N w() =, w () = γ >, which ha a unique olution for every γ >, and find a poitive olution in (, b a). Oberve that olution to (2.4) verify on one hand w cr 2(N ) 2, o that an integration provide w() (γ cr 2(N ) 2 /2). On the other hand, ince w i decreaing we have w ( ) = for ome >, and it follow by the ymmetry of the problem that w i ymmetric with repect to and w(2 ) =. Letting 2 be the firt zero of w and integrating the equation in (, ) we obtain ( ) N γ R R N = R 2 2 dt g(t) + c. We conclude that i an increaing function of γ which verifie ( ) N R dt (2.5) g(t) + c R 2 2 a γ +. Therefore in the cae where (2.2) hol, ince g i increaing, then the integral in (2.5) diverge. So, we can alway chooe γ large enough o that > (b a)/2, and thi provide with a poitive olution of (2.). When the integral converge, we can alo obtain > (b a)/2 if we elect c mall enough, ince the integral (2.6) dt g(t) diverge at, due to g() = and g C (R). Thi conclude the proof. Lemma 4. Aume g C (R) i increaing with g() = and α >. Then, if (2.7) =, g() for every c > there exit a poitive radial function u verifying (2.). If (2.7) doe not hold and c i mall enough, uch a function alo exit. Proof. Setting z = c/α u, we look for a function verifying (r N z ) r N (αz + g( z )) z(r ) = c α, z(r 2) c α.
6 6 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS We will look for a poitive olution z to thi inequality. With the change of variable (2.3), and letting v() = z(r), we find a before that v i a uperolution provided for intance that v R 2(N ) 2 (αv + g( R N v )) v(a) = c, v(b) =. α Setting w() = v(b ), it i thu natural to conider the initial value problem (2.8) w = R 2(N ) 2 ( αw + g ( w() =, w () = γ >, R N )) w which ha a unique olution for every γ >, and ee if we can elect γ o that w(b a) i a large a we pleae. Notice that w a long a w, o that it i not hard to ee that olution are poitive, increaing and convex for >. For every γ >, the olution i defined in an interval [, T (γ)), and when T (γ) < we have (2.9) lim w() = + or lim T (γ) T (γ) w () = +. Let u ee that when the integral condition (2.7) i atified, we alway have both condition in (2.9). Indeed, the firt one implie the econd, and if we had w(t (γ)) < +, then ( ( w R 2(N ) 2 αw(t (γ)) + g Multiplying by w and integrating we arrive at which yiel T w w αw(t (γ)) + g( γ R N R N αw(t (γ)) + g() R N w )) w ) R2(N ) 2 w(t (γ)), R2(N ) 2 R N w(t (γ)), contradicting condition (2.7). Thu w, w a T (γ). We have two cae to conider: either T (γ ) i infinite for ome γ > or T (γ) i finite for every γ >. In the firt cae, let u ee that thi implie T (γ) = for every γ >. Oberve firt that when u, v are two olution to the equation in (2.8) with u() v(), u () = γ > γ 2 = v (), then u > v in the common interval of definition, hence T (γ ) T (γ 2 ). In particular, T (γ) = for γ < γ. If γ > γ and we temporarily denote by w γ the unique olution to (2.8), there exit δ > uch that w γ (δ) > γ, ince w γ i increaing and converge to infinity. Let w(x) = w γ (x + δ). Then w i a olution to the ame equation with initial data w() = w γ (δ) >, w () = w γ (δ) > γ. It follow by the previou obervation that w > w γ, and in particular T (γ) =. Thu all olution are global in thi cae and it i eay to conclude: ince w(x) γx by convexity, we can have w(b a).
7 GENERAL CONVECTION TERM 7 a large a we pleae, o that a uperolution can be contructed with large value of c. The econd poibility i that all olution blowup in finite time, i. e. T (γ) < for every γ >. Let u ee that in uch cae T (γ) i a continuou function of γ. Take γ n γ. By comparion we have T (γ n ) < T (γ). Moreover, we can chooe δ n uch that w γ(δ n ) > γ n. Arguing a before, w γ (x+δ n ) > w γn (x), o that T (γ) δ n < T (γ n ) and we obtain T (γ n ) T (γ). When γ n γ the proof i imilar. Next, we claim that T (γ) a γ. Indeed, aume T (γ) T when γ. Since w, we obtain w T w, o that ( ( w R 2(N ) 2 γ R N g R N w ) + αt w ) and thi lea, after an integration and a change of variable, to ( ) R 2 N g() + αt R N 2 T. R A contradiction i reached when we let γ, ince the integral then diverge. Thu lim γ T (γ) =. Let u denote T = lim γ T (γ) (which i expected to be zero). If T b a we can ue the continuity of T to obtain γ > uch that T (γ) = b a+ε for mall poitive ε. Taking ε a mall a we pleae we obtain w γ (b a) a large a we wih, and thi provide with a uperolution for large value of c. If, on the contrary, T > b a, then all olution would be defined at leat in [, b a] and ince w(b a) γ(b a), we obtain that w γ (b a) i a large a we pleae by taking large value of γ. To conclude the proof, we now conider the cae when g() < and c i mall enough. Oberve that in thi cae all olution blow up in finite time. Indeed, let T < T (γ). Since ( ) w R 2(N ) 2 g w R N we can integrate in (, T ) and let T T (γ) to arrive at: (2.) R 2(N ) 2 T (γ) R N γ g() <, ince thi lat integral alo converge. It alo follow from (2.) that T (γ) a γ (i. e. T = in the above proof). Since T (γ) i continuou with T (γ) a γ, we can chooe γ uch that T (γ) > b a and obtain a uperolution for c αw γ (T (γ)). It i worth mentioning that in the preent cae where (2.7) doe not hold we cannot guarantee that the firt equality in (2.9) hol, o that the uperolution i not valid in principle for large value of c.,
8 8 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS 3. Nonexitence of olution in ball We tackle in thi ection the quetion of nonexitence of olution to (.). We will ee in Section 4 that it uffice to how nonexitence of radial olution when Ω i a ball of R N and h i contant. Thu, under everal hypothee, we will how that the problem u N u = αu + g( u ) + c, < r < R (3.) r u () =, u(r) = doe not admit poitive olution for large value of c. Lemma 5. Aume g C (R) i increaing with g() = and α =. Then if g() <, there exit c > uch that problem (3.) doe not admit poitive olution when c c. Proof. Aume u i a olution to (3.). We firt claim that u (r) < for r (, R) and u (r) < in [, R). Oberve that u () = c/n <, o that u (r) < for r > cloe enough to zero. If we had u (r ) = for ome r (, R) with u < in (, r ), then u (r ) o that from the equation we obtain u (r ) = c <, which i impoible. Then u (r) < if < r < R. Aume now that for ome r (, R) we have u ( r ) =. Since u ( r ) in thi cae, we obtain by differentiating the equation u N u + N r r 2 u = g ( u )u o that u ( r ) <, a contradiction. Thu u (r) < for r (, R) a well. Next if we rewrite the equation a (r N u ) = r N (g( u ) + c) and integrate in (, r) we obtain, taking into account that g( u ) i increaing: r N u (r) = r N (g( u ()) + c) (g( u (r)) + c) o that plugging thi in (3.) we have r N = rn N (g( u (r)) + c), u N (g( u ) + c) in (, R). Integrating in (, R) we obtain dt g(t) + c > u (R) dt g(t) + c N R. Thi implie that c cannot be too large in order to have a poitive olution to (3.).
9 GENERAL CONVECTION TERM 9 Lemma 6. Aume g C (R) i increaing with g() = and α >. Then if (3.2) <, g() there exit c > uch that problem (3.) doe not admit poitive olution when c c. Proof. Let u be a poitive olution to (3.). We firt claim that u < c/α. Indeed, if we had u() = c/α then u c/α by uniquene, which i not poible. If u() > c/α then u () > and u initially increae. According to the boundary condition u(r) =, there hould be a point where u achieve it maximum, but thi i in contradiction with the equation. We conclude that u() < c/α and again by the equation u initially decreae and cannot reach a minimum, o u i alway decreaing. It i een much a in the previou cae that u < in [, R) alo. Thu arguing a in that proof we obtain (3.3) u N ( αu + g( u ) + c). Aume there exit a equence c n uch that a poitive olution u n to (3.) exit with c = c n (with no lo of generality we may aume that c n i increaing). Let v n = c n α u n. Then v n + N v n = αv n + g(v r n) v n() =, v n (R) = c n α, with v n >, v n >. We claim that v n () i bounded a n. Indeed, ince from (3.3) we have we can integrate to arrive at v n N (αv n + g(v n)) N (αv n() + g(v n)), N R v n (R) αv n () + g() < αv n () + g(). Therefore if v n () we arrive at a contradiction. Since olution are increaing in c (thank to uniquene), we can guarantee that v n () v for ome v >. It alo follow that v n z, the unique olution to z + N z = αz + g(z ) r z() = v, z () =, which i defined in a maximal interval [, T ). When T <, we have lim r T z(r) = or lim r T z (r) =. By comparion we alo have v n z in [, mint, R}). Let u ee that T < R. Indeed, if T R, we would have v n (R) = c nα z(r), and then T = R, z(r) = follow. Thi i impoible, ince
10 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS z N g(z ), and multiplication by z and another integration between R 2 and R ε for ome mall poitive ε yiel N (z(r ε) z(r 2 )) z (R ε) z (R/2) g() < z (R/2) g(). Letting ε we obtain a contradiction with (3.2). Thu T < R. Now chooe a mall ε >. Since v n z uniformly in [, T ε], we have v n(t ε) z (T ε) ε if n i large enough. Therefore (R T + ε) N v (R) v n(t ε) g() < z (T ε) ε g(). Letting ε, we arrive at T R, a contradiction which how that no olution to (3.) may exit if c i large enough. 4. Proof of the theorem Thi final ection will be dedicated to the proof of Theorem and 2. The idea of the proof of exitence come from [24], and it conit in truncating the function g in order to obtain a olution, and then etimating the gradient of thi olution in Ω. The eential point i to obtain a uitable uperolution. Since Ω verifie the uniform exterior phere condition, there exit R > uch that for every x Ω there exit y R N \Ω with B R (y ) Ω = x }. Chooe R 2 > R large enough o that Ω A := B R2 (y ) \ B R (y ) for every x Ω. Conider the radial problem (r N u ) = r N ( αu + g( u ) + c), R < r < R 2 (4.) u(r ) =, u(r 2 ), where c >. The firt important exitence reult i the following: Lemma 7. Aume g C (R) i increaing with g() = and h C (Ω) i nonnegative. If there exit a poitive uperolution ū to problem (4.), then for every λ (, c/ h ], problem (.) admit a poitive olution. With regard to nonexitence reult, the reference ituation i a radial problem in a ball. Oberve that, ince Ω verifie a uniform interior ball condition and h, h on Ω, we can find x Ω, y Ω and R > uch that B R (y ) Ω, B R (y ) Ω = x } and h h > in B R (y ). Conider the problem (r N u ) = r N ( αu + g( u ) + c), < r < R (4.2) u () =, u(r) =, where c >. Then: Lemma 8. Aume g C (R) i increaing with g() = and h C (Ω) i nonnegative with h h > in B R (y ). If problem (4.2) doe not admit a poitive olution for ome c >, then problem (.) doe not have olution for λ c/h. A we have quoted in the introduction, the method we follow for proving exitence (and indeed alo nonexitence) relie in obtaining good etimate for the gradient of the olution. Thi lat part i achieved by mean of a
11 GENERAL CONVECTION TERM kind of maximum principle for the gradient of olution to (.). The proof i inpired in the claical method of Berntein (ee for intance [29] or [24]). Lemma 9. Let u C 2 (Ω) C (Ω) be a olution to (.). Aume g C (R) i increaing with g() = and h C (Ω). Then there exit a contant C which depen on up Ω u, up Ω u, up Ω h and λ uch that u C in Ω. Proof. Let u be a olution to (.), and define w = u 2 +u 2. For implicity, let u denote g( ξ ) = g( ξ 2 ). By tandard regularity, it follow that u C 3 (Ω ρ ), where Ω ρ = x Ω : u 2 > ρ} for ome < ρ < u, and hence w C 2 (Ω ρ ) C(Ω ρ ). Then, it i not hard to check that in Ω ρ one ha w = 2 N D2 u 2 2 g ( u 2 ) u (w u 2 ) 2λ h u+2u u+(2+2α) u 2. On the other hand, ( N 2 ( u) 2 = ii u) N i= N ( ii u) 2 N D 2 u 2, and ince g i nondecreaing and u, o that 2 g ( u 2 ) u (u 2 ), we have w 2 N ( u)2 2 g ( u 2 ) u w 2λ h u + 2u u + (2 + 2α) u 2 in Ω ρ. An application of CauchySchwarz inequality lea to w N ( u)2 2 g ( u 2 ) u w λ 2 h 2 Nu 2 + u 2 in Ω ρ. Fix M > up u 2 + 2N u 2 + λ 2 h, Ω and aume that the open et Ω = x Ω : w > M} i nonempty. It clearly follow that Ω Ω ρ ince i= u 2 u 2 > ρ in Ω. Hence Lw in Ω, where Lw := w 2g ( u ) u w, and the trong maximum principle implie w < up Ω w = M in Ω, a contradiction. Hence w M in Ω. Now we come to the proof of Lemma 7 and 8. Proof of Lemma 7. Fix for the moment x Ω and denote v(x) = ū( x y ). Take K > and let g K C (R) be a bounded increaing function verifying g K (t) = g(t) if t K. Let u conider the truncated problem u + αu = gk ( u ) + λh(x) in Ω (4.3) u = on Ω. When K > up ū, the function v i a uperolution to (4.3) and ince v = a ubolution, it follow that there exit a olution u to (4.3) (by uing the reult in [6] or [28]), which verifie u v.
12 2 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS By the maximum principle we have u > in Ω and u ν Moreover, ince v(x ) =, < on Ω. u ν (x ) v ν (x ) = ū (R ). Let u ee that the ame inequality hol for every x Ω. Indeed, if we take uch a x and A i the correponding annulu, then ince u + αu g K ( u ) + λ h in Ω, the function ū( x y ) i a uperolution to the problem (4.3), conidered in A. By comparion we obtain u(x) ū( x y ) in Ω. Thu u ν (x ) ū (R ). Hence ū (R ) u ν on Ω. We are in a poition to apply Lemma 9 to obtain a contant M >, which doe not depend on K, uch that u < M in Ω. Taking K > M, we have g K ( u ) = g( u ) in Ω and u i a olution to our original problem. Thi conclude the proof. Proof of Lemma 8. Aume problem (.) ha a poitive olution u for ome λ c/h. Then u i a uperolution to the problem v + αv = g( v ) + λh in B R (y ) v = on B R (y ). A imilar procedure a in Lemma 7 yiel the exitence of a olution to thi problem, which i unique, hence radial. Thi i in contradiction with the hypothei. It i important to remark that thi procedure work ince the uperolution vanihe at x B R (y ) Ω, which allow u to etimate v on B R (y ) in term of u(x ). Finally, we proceed to prove our main theorem. We notice that, once we have analyzed eparately the cae α = and α > in Section 2 and 3, the ret of the proof i exactly the ame in both cae. Proof of Theorem and 2. (i) By Lemma 3 and 4, there exit a uperolution to (4.) for every c >. The exitence of a poitive olution to (.) for every λ > follow thank to Lemma 7. (ii) Again by lemma 3, 4 and 7, there exit a olution for mall value of λ. On the other hand, uing Lemma 8 in conjunction with Lemma 5 and 6 we alo have that no olution to (.) exit for large value of λ. Hence, we can define Λ = upλ > : there exit a olution to (.)} and Λ i finite and poitive. By it very definition, there are no olution to (.) for λ > Λ. Now, chooe an arbitrary λ (, Λ). Then there exit µ (λ, Λ) uch that (.) with λ ubtituted by µ admit a poitive olution v. Since thi olution i a uperolution to (.), the exitence of a poitive olution follow a in Lemma 7.
13 GENERAL CONVECTION TERM 3 Acknowledgement. S. A. wa partially upported by USM Grant No. 22, J. GM wa upported by Miniterio de Ciencia e Innovación and FEDER under grant MTM (Spain) and A. Q. wa partially upported by Fondecyt Grant No. 2 and CAPDE, Anillo ACT25. All three author were partially upported by Programa Baal CMM, U. de Chile. Reference [] H. Abdel Hamid, M. F. BidautVeron, Correlation between two quailinear elliptic problem with a ource term involving the function or it gradient, C. R. Acad. Sci. Pari Ser. I Math. 346 (28), [2] B. Abdellaoui, A. Dall Aglio, I. Peral, Some remark on elliptic problem with critical growth in the gradient, J. Diff. Equation 222, (26), [3] N. Alaa, M. Pierre, Weak olution of ome quailinear elliptic equation with data meaure, SIAM J. Math. Anal. 24 (993), [4] S. Alarcón, J. GarcíaMelián, A. Quaa, KellerOerman type condition for ome elliptic problem with gradient term, to appear in J. Diff. Eqn. [5] A. Alvino, G. Trombetti, P.L. Lion, Comparion reult for elliptic and parabolic equation via Schwarz ymmetrization, Ann. Int. H. Poincaré Anal. Non Linaire 7 (99), [6] H. Amann, M. G. Crandall, On ome exitence theorem for emilinear elliptic equation, Indiana Univ. Math. J. 27 (978), [7] G. Barle, A hort proof of the C,α regularity of vicoity ubolution for uperquadratic vicou HamiltonJacobi equation and application, Nonlinear Anal. 73 (2), [8] G. Barle, A. P. Blanc, C. Georgelin, M. Kobylanki, Remark on the maximum principle for nonlinear elliptic PDE with quadratic growth condition, Ann. Scuola Norm. Sup. Pia 28 (999), [9] G. Barle, F. Murat, Uniquene and the maximum principle for quailinear elliptic equation with quadratic growth condition, Arch. Rational. Mech Anal. 33 (995), 77. [] G. Barle, A. Porretta, Uniquene for unbounded olution to tationary vicou HamiltonJacobi equation, Ann. Sc. Norm. Super. Pia Cl. Sci. 5 (26), [] L. Boccardo, T. Gallouët, F. Murat, A unified preentation of two exitence reult for problem with natural growth, in Progre in partial differential equation: the Metz urvey, 2 (992), 27 37, Pitman Re. Note Math. Ser. 296, Longman Sci. Tech., Harlow, 993. [2] L. Boccardo, F. Murat, J. P. Puel, Reultat d exitence pour certain probleme elliptique quailineaire, Ann. Scuola Norm. Sup. Pia, (984), [3] L. Boccardo, F. Murat, J. P. Puel, L etimate for ome nonlinear elliptic partial differential equation and application to an exitence reult, SIAM J. Math. Anal. 23 (2) (992), [4] A. Dall Aglio, D. Giachetti, J. P. Puel, Nonlinear elliptic equation with natural growth in general domain, Ann. Mat. Pura Appl. 8 (22), [5] T. Del Vecchio, M.M. Porzio, Exitence reult for a cla of noncoercive Dirichlet problem, Ricerche Mat. 44 (995), [6] P. Felmer, A. Quaa, On the trong maximum principle for quailinear elliptic equation and ytem, Adv. Differential Equation 7 (22), no., [7] V. Ferone, F. Murat, Quailinear problem having quadratic growth in the gradient: an exitence reult when the ource term i mall, Equation aux derivee partielle et application, GauthierVillar, Ed. Sci. Med. Elevier, Pari (998), [8] V. Ferone, F. Murat, Nonlinear problem having quadratic growth in the gradient: an exitence reult when the ource term i mall, Non. Anal. TMA 42 (2),
14 4 S. ALARCÓN, J. GARCÍAMELIÁN AND A. QUAAS [9] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equation of econd order, Springer Verlag, 983. [2] N. Grenon, F. Murat, A. Porretta, Exitence and a priori etimate for elliptic problem with ubquadratic gradient dependent term, C. R. Math. Acad. Sci. Pari 342 (), (26), [2] K. Hanon, V. Mazja, I.E. Verbitky, Criteria of olvability for multidimenional Riccati equation, Ark. Mat. 37 (999), [22] L. Jeanjean, B. Sirakov, Multiple olution for an elliptic problem with quadratic growth in the gradient, preprint. [23] J. M. Lary, P. L. Lion, Nonlinear elliptic equation with ingular boundary condition and tochatic control with tate contraint. I. The model problem, Math. Ann. 283 (989), [24] P. L. Lion, Réolution de problème elliptique quailinéaire, Arch. Rat. Mech. Anal. 74 (98), [25] P. L. Lion, Quelque remarque ur le probleme elliptique quailineaire du econd ordre, J. Anal. Math. 45 (985), [26] A. Porretta, The ergodic limit for a vicou HamiltonJacobi equation with Dirichlet condition, Rend. Lincei (9) Mat. Appl. 2 (2), [27] P. Pucci, J. Serrin, H. Zou, A trong maximum principle and a compact upport principle for ingular elliptic equation, J. Math. Pure Appl. 78 (999), [28] J. SchoenenbergerDeuel, P. He, A criterion for the exitence of olution of nonlinear elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A 74 (974/75), (976). [29] J. Serrin, The problem of Dirichlet for quailinear elliptic differential equation with many independent variable, Philo. Tran. R. Soc. London A 264 (969), [3] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Rat. Mech. Anal. 95 (2), S. Alarcón and A. Quaa Departamento de Matemática, Univeridad Técnica Federico Santa María Cailla V, Avda. Epaña, 68 Valparaío, CHILE. addre: J. GarcíaMelián Departamento de Análii Matemático, Univeridad de La Laguna. C/. Atrofíico Francico Sánchez /n, 3827 La Laguna, SPAIN and Intituto Univeritario de Etudio Avanzado (IUdEA) en Fíica Atómica, Molecular y Fotónica, Facultad de Fíica, Univeridad de La Laguna C/. Atrofíico Francico Sánchez /n, 3823 La Laguna, SPAIN addre:
Unit 11 Using Linear Regression to Describe Relationships
Unit 11 Uing Linear Regreion to Decribe Relationhip Objective: To obtain and interpret the lope and intercept of the leat quare line for predicting a quantitative repone variable from a quantitative explanatory
More informationQueueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems,
MANAGEMENT SCIENCE Vol. 54, No. 3, March 28, pp. 565 572 in 25199 ein 1526551 8 543 565 inform doi 1.1287/mnc.17.82 28 INFORMS Scheduling Arrival to Queue: A SingleServer Model with NoShow INFORMS
More informationv = x t = x 2 x 1 t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled t
Chapter 2 Motion in One Dimenion 2.1 The Important Stuff 2.1.1 Poition, Time and Diplacement We begin our tudy of motion by conidering object which are very mall in comparion to the ize of their movement
More informationA note on profit maximization and monotonicity for inbound call centers
A note on profit maximization and monotonicity for inbound call center Ger Koole & Aue Pot Department of Mathematic, Vrije Univeriteit Amterdam, The Netherland 23rd December 2005 Abtract We conider an
More informationPartial optimal labeling search for a NPhard subclass of (max,+) problems
Partial optimal labeling earch for a NPhard ubcla of (max,+) problem Ivan Kovtun International Reearch and Training Center of Information Technologie and Sytem, Kiev, Uraine, ovtun@image.iev.ua Dreden
More informationAssessing the Discriminatory Power of Credit Scores
Aeing the Dicriminatory Power of Credit Score Holger Kraft 1, Gerald Kroiandt 1, Marlene Müller 1,2 1 Fraunhofer Intitut für Techno und Wirtchaftmathematik (ITWM) GottliebDaimlerStr. 49, 67663 Kaierlautern,
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationA Note on Profit Maximization and Monotonicity for Inbound Call Centers
OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1304 1308 in 0030364X ein 15265463 11 5905 1304 http://dx.doi.org/10.1287/opre.1110.0990 2011 INFORMS TECHNICAL NOTE INFORMS hold copyright
More informationProject Management Basics
Project Management Baic A Guide to undertanding the baic component of effective project management and the key to ucce 1 Content 1.0 Who hould read thi Guide... 3 1.1 Overview... 3 1.2 Project Management
More informationOptical Illusion. Sara Bolouki, Roger Grosse, Honglak Lee, Andrew Ng
Optical Illuion Sara Bolouki, Roger Groe, Honglak Lee, Andrew Ng. Introduction The goal of thi proect i to explain ome of the illuory phenomena uing pare coding and whitening model. Intead of the pare
More informationTwo Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL
Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy
More informationMixed Method of Model Reduction for Uncertain Systems
SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced
More informationTIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME
TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME RADMILA KOCURKOVÁ Sileian Univerity in Opava School of Buine Adminitration in Karviná Department of Mathematical Method in Economic Czech Republic
More informationA technical guide to 2014 key stage 2 to key stage 4 value added measures
A technical guide to 2014 key tage 2 to key tage 4 value added meaure CONTENTS Introduction: PAGE NO. What i value added? 2 Change to value added methodology in 2014 4 Interpretation: Interpreting chool
More informationSenior Thesis. Horse Play. Optimal Wagers and the Kelly Criterion. Author: Courtney Kempton. Supervisor: Professor Jim Morrow
Senior Thei Hore Play Optimal Wager and the Kelly Criterion Author: Courtney Kempton Supervior: Profeor Jim Morrow June 7, 20 Introduction The fundamental problem in gambling i to find betting opportunitie
More informationDMA Departamento de Matemática e Aplicações Universidade do Minho
Univeridade do Minho DMA Departamento de Matemática e Aplicaçõe Univeridade do Minho Campu de Gualtar 4757 Braga Portugal www.math.uminho.pt Univeridade do Minho Ecola de Ciência Departamento de Matemática
More informationRedesigning Ratings: Assessing the Discriminatory Power of Credit Scores under Censoring
Redeigning Rating: Aeing the Dicriminatory Power of Credit Score under Cenoring Holger Kraft, Gerald Kroiandt, Marlene Müller Fraunhofer Intitut für Techno und Wirtchaftmathematik (ITWM) Thi verion: June
More informationMECH 2110  Statics & Dynamics
Chapter D Problem 3 Solution 1/7/8 1:8 PM MECH 11  Static & Dynamic Chapter D Problem 3 Solution Page 7, Engineering Mechanic  Dynamic, 4th Edition, Meriam and Kraige Given: Particle moving along a traight
More informationQueueing Models for Multiclass Call Centers with RealTime Anticipated Delays
Queueing Model for Multicla Call Center with RealTime Anticipated Delay Oualid Jouini Yve Dallery Zeynep Akşin Ecole Centrale Pari Koç Univerity Laboratoire Génie Indutriel College of Adminitrative Science
More informationOnline story scheduling in web advertising
Online tory cheduling in web advertiing Anirban Dagupta Arpita Ghoh Hamid Nazerzadeh Prabhakar Raghavan Abtract We tudy an online job cheduling problem motivated by toryboarding in web advertiing, where
More informationMSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring 2013. Contents
MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market Anne Sibert Revied Spring 203 Content Introduction................................................. 2 The Mone Market.............................................
More informationLinear energypreserving integrators for Poisson systems
BIT manucript No. (will be inerted by the editor Linear energypreerving integrator for Poion ytem David Cohen Ernt Hairer Received: date / Accepted: date Abtract For Hamiltonian ytem with noncanonical
More informationChapter 10 Stocks and Their Valuation ANSWERS TO ENDOFCHAPTER QUESTIONS
Chapter Stoc and Their Valuation ANSWERS TO ENOFCHAPTER QUESTIONS  a. A proxy i a document giving one peron the authority to act for another, typically the power to vote hare of common toc. If earning
More informationSocially Optimal Pricing of Cloud Computing Resources
Socially Optimal Pricing of Cloud Computing Reource Ihai Menache Microoft Reearch New England Cambridge, MA 02142 timena@microoft.com Auman Ozdaglar Laboratory for Information and Deciion Sytem Maachuett
More informationControl of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling
Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada
More informationHUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * Michael Spagat Royal Holloway, University of London, CEPR and Davidson Institute.
HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * By Michael Spagat Royal Holloway, Univerity of London, CEPR and Davidon Intitute Abtract Tranition economie have an initial condition of high human
More informationBidding for Representative Allocations for Display Advertising
Bidding for Repreentative Allocation for Diplay Advertiing Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii Yahoo! Reearch. {arpita, mcafee, kpapi, ergei}@yahooinc.com Abtract. Diplay
More informationChapter 32. OPTICAL IMAGES 32.1 Mirrors
Chapter 32 OPTICAL IMAGES 32.1 Mirror The point P i called the image or the virtual image of P (light doe not emanate from it) The leftright reveral in the mirror i alo called the depth inverion (the
More information1) Assume that the sample is an SRS. The problem state that the subjects were randomly selected.
12.1 Homework for t Hypothei Tet 1) Below are the etimate of the daily intake of calcium in milligram for 38 randomly elected women between the age of 18 and 24 year who agreed to participate in a tudy
More informationProfitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations
Proceeding of the 0 Indutrial Engineering Reearch Conference T. Doolen and E. Van Aken, ed. Profitability of Loyalty Program in the Preence of Uncertainty in Cutomer Valuation Amir Gandomi and Saeed Zolfaghari
More informationOriginal Article: TOWARDS FLUID DYNAMICS EQUATIONS
Peer Reviewed, Open Acce, Free Online Journal Publihed monthly : ISSN: 88X Iue 4(5); April 15 Original Article: TOWARDS FLUID DYNAMICS EQUATIONS Citation Zaytev M.L., Akkerman V.B., Toward Fluid Dynamic
More informationMath 22B, Homework #8 1. y 5y + 6y = 2e t
Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.
More informationA Resolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networks
A Reolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networ Joé Craveirinha a,c, Rita GirãoSilva a,c, João Clímaco b,c, Lúcia Martin a,c a b c DEECFCTUC FEUC INESCCoimbra International
More informationName: SID: Instructions
CS168 Fall 2014 Homework 1 Aigned: Wedneday, 10 September 2014 Due: Monday, 22 September 2014 Name: SID: Dicuion Section (Day/Time): Intruction  Submit thi homework uing Pandagrader/GradeScope(http://www.gradecope.com/
More informationLinear Momentum and Collisions
Chapter 7 Linear Momentum and Colliion 7.1 The Important Stuff 7.1.1 Linear Momentum The linear momentum of a particle with ma m moving with velocity v i defined a p = mv (7.1) Linear momentum i a vector.
More informationSCM integration: organiational, managerial and technological iue M. Caridi 1 and A. Sianei 2 Dipartimento di Economia e Produzione, Politecnico di Milano, Italy Email: maria.caridi@polimi.it Itituto
More informationSolutions to Sample Problems for Test 3
22 Differential Equation Intructor: Petronela Radu November 8 25 Solution to Sample Problem for Tet 3 For each of the linear ytem below find an interval in which the general olution i defined (a) x = x
More informationHealth Insurance and Social Welfare. Run Liang. China Center for Economic Research, Peking University, Beijing 100871, China,
Health Inurance and Social Welfare Run Liang China Center for Economic Reearch, Peking Univerity, Beijing 100871, China, Email: rliang@ccer.edu.cn and Hao Wang China Center for Economic Reearch, Peking
More informationNETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET
Chapter 1 NETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET S. Srivatava Univerity of Miouri Kana City, USA hekhar@conrel.ice.umkc.edu S. R. Thirumalaetty now
More informationGroup Mutual Exclusion Based on Priorities
Group Mutual Excluion Baed on Prioritie Karina M. Cenci Laboratorio de Invetigación en Sitema Ditribuido Univeridad Nacional del Sur Bahía Blanca, Argentina kmc@c.un.edu.ar and Jorge R. Ardenghi Laboratorio
More informationCASE STUDY BRIDGE. www.futureprocessing.com
CASE STUDY BRIDGE TABLE OF CONTENTS #1 ABOUT THE CLIENT 3 #2 ABOUT THE PROJECT 4 #3 OUR ROLE 5 #4 RESULT OF OUR COLLABORATION 67 #5 THE BUSINESS PROBLEM THAT WE SOLVED 8 #6 CHALLENGES 9 #7 VISUAL IDENTIFICATION
More informationScheduling of Jobs and Maintenance Activities on Parallel Machines
Scheduling of Job and Maintenance Activitie on Parallel Machine ChungYee Lee* Department of Indutrial Engineering Texa A&M Univerity College Station, TX 778433131 cylee@ac.tamu.edu ZhiLong Chen** Department
More informationSupport Vector Machine Based Electricity Price Forecasting For Electricity Markets utilising Projected Assessment of System Adequacy Data.
The Sixth International Power Engineering Conference (IPEC23, 2729 November 23, Singapore Support Vector Machine Baed Electricity Price Forecating For Electricity Maret utiliing Projected Aement of Sytem
More informationThus far. Inferences When Comparing Two Means. Testing differences between two means or proportions
Inference When Comparing Two Mean Dr. Tom Ilvento FREC 48 Thu far We have made an inference from a ingle ample mean and proportion to a population, uing The ample mean (or proportion) The ample tandard
More informationReal Business Cycles. Jesus FernandezVillaverde University of Pennsylvania
Real Buine Cycle Jeu FernandezVillaverde Univerity of Pennylvania 1 Buine Cycle U.S. economy uctuate over time. How can we build model to think about it? Do we need dierent model than before to do o?
More informationTRADING rules are widely used in financial market as
Complex Stock Trading Strategy Baed on Particle Swarm Optimization Fei Wang, Philip L.H. Yu and David W. Cheung Abtract Trading rule have been utilized in the tock market to make profit for more than a
More informationCASE STUDY ALLOCATE SOFTWARE
CASE STUDY ALLOCATE SOFTWARE allocate caetud y TABLE OF CONTENTS #1 ABOUT THE CLIENT #2 OUR ROLE #3 EFFECTS OF OUR COOPERATION #4 BUSINESS PROBLEM THAT WE SOLVED #5 CHALLENGES #6 WORKING IN SCRUM #7 WHAT
More informationRisk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms
Rik Management for a Global Supply Chain Planning under Uncertainty: Model and Algorithm Fengqi You 1, John M. Waick 2, Ignacio E. Gromann 1* 1 Dept. of Chemical Engineering, Carnegie Mellon Univerity,
More informationEfficient Pricing and Insurance Coverage in Pharmaceutical Industry when the Ability to Pay Matters
ömmföäfläafaäflaflafla fffffffffffffffffffffffffffffffffff Dicuion Paper Efficient Pricing and Inurance Coverage in Pharmaceutical Indutry when the Ability to Pay Matter Vea Kanniainen Univerity of Helinki,
More informationSolution of the Heat Equation for transient conduction by LaPlace Transform
Solution of the Heat Equation for tranient conduction by LaPlace Tranform Thi notebook ha been written in Mathematica by Mark J. McCready Profeor and Chair of Chemical Engineering Univerity of Notre Dame
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationGrowth and Sustainability of Managed Security Services Networks: An Economic Perspective
Growth and Sutainability of Managed Security Service etwork: An Economic Perpective Alok Gupta Dmitry Zhdanov Department of Information and Deciion Science Univerity of Minneota Minneapoli, M 55455 (agupta,
More information2. METHOD DATA COLLECTION
Key to learning in pecific ubject area of engineering education an example from electrical engineering AnnaKarin Cartenen,, and Jonte Bernhard, School of Engineering, Jönköping Univerity, S Jönköping,
More informationAuction Theory. Jonathan Levin. October 2004
Auction Theory Jonathan Levin October 2004 Our next topic i auction. Our objective will be to cover a few of the main idea and highlight. Auction theory can be approached from different angle from the
More informationManaging Customer Arrivals in Service Systems with Multiple Servers
Managing Cutomer Arrival in Service Sytem with Multiple Server Chrito Zacharia Department of Management Science, School of Buine Adminitration, Univerity of Miami, Coral Gable, FL 3346. czacharia@bu.miami.edu
More informationJanuary 21, 2015. Abstract
T S U I I E P : T R M C S J. R January 21, 2015 Abtract Thi paper evaluate the trategic behavior of a monopolit to influence environmental policy, either with taxe or with tandard, comparing two alternative
More informationIntroduction to the article Degrees of Freedom.
Introduction to the article Degree of Freedom. The article by Walker, H. W. Degree of Freedom. Journal of Educational Pychology. 3(4) (940) 5369, wa trancribed from the original by Chri Olen, George Wahington
More informationDISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS. G. Chapman J. Cleese E. Idle
DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS G. Chapman J. Cleee E. Idle ABSTRACT Content matching i a neceary component of any ignaturebaed network Intruion Detection
More informationResearch Article An (s, S) Production Inventory Controlled SelfService Queuing System
Probability and Statitic Volume 5, Article ID 558, 8 page http://dxdoiorg/55/5/558 Reearch Article An (, S) Production Inventory Controlled SelfService Queuing Sytem Anoop N Nair and M J Jacob Department
More informationReport 46681b 30.10.2010. Measurement report. Sylomer  field test
Report 46681b Meaurement report Sylomer  field tet Report 46681b 2(16) Contet 1 Introduction... 3 1.1 Cutomer... 3 1.2 The ite and purpoe of the meaurement... 3 2 Meaurement... 6 2.1 Attenuation of
More informationTowards ControlRelevant Forecasting in Supply Chain Management
25 American Control Conference June 81, 25. Portland, OR, USA WeA7.1 Toward ControlRelevant Forecating in Supply Chain Management Jay D. Schwartz, Daniel E. Rivera 1, and Karl G. Kempf Control Sytem
More informationFEDERATION OF ARAB SCIENTIFIC RESEARCH COUNCILS
Aignment Report RP/98983/5/0./03 Etablihment of cientific and technological information ervice for economic and ocial development FOR INTERNAL UE NOT FOR GENERAL DITRIBUTION FEDERATION OF ARAB CIENTIFIC
More informationLECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS
LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS. Complete bipartite graph.. KövariSóTurán theorem. When H i a bipartite graph, i.e., when χ(h) = 2, Erdő and Stone theorem aert that π H = 0. In other word,
More informationTurbulent Mixing and Chemical Reaction in Stirred Tanks
Turbulent Mixing and Chemical Reaction in Stirred Tank André Bakker Julian B. Faano Blend time and chemical product ditribution in turbulent agitated veel can be predicted with the aid of Computational
More informationOn the absolute continuity of onedimensional SDE s driven by a fractional Brownian motion
On the abolute continuity of onedimenional SDE driven by a fractional Brownian motion Ivan Nourdin Univerité Henri Poincaré, Intitut de Mathématique Élie Cartan, B.P. 239 5456 VandœuvrelèNancy Cédex,
More informationReview of Multiple Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015
Review of Multiple Regreion Richard William, Univerity of Notre Dame, http://www3.nd.edu/~rwilliam/ Lat revied January 13, 015 Aumption about prior nowledge. Thi handout attempt to ummarize and yntheize
More informationHOMOTOPY PERTURBATION METHOD FOR SOLVING A MODEL FOR HIV INFECTION OF CD4 + T CELLS
İtanbul icaret Üniveritei Fen Bilimleri Dergii Yıl: 6 Sayı: Güz 7/. 95 HOMOOPY PERURBAION MEHOD FOR SOLVING A MODEL FOR HIV INFECION OF CD4 + CELLS Mehmet MERDAN ABSRAC In thi article, homotopy perturbation
More informationFinite Automata. a) Reading a symbol, b) Transferring to a new instruction, and c) Advancing the tape head one square to the right.
Finite Automata Let u begin by removing almot all of the Turing machine' power! Maybe then we hall have olvable deciion problem and till be able to accomplih ome computational tak. Alo, we might be able
More informationNewton s Laws. A force is simply a push or a pull. Forces are vectors; they have both size and direction.
Newton Law Newton firt law: An object will tay at ret or in a tate of uniform motion with contant velocity, in a traight line, unle acted upon by an external force. In other word, the bodie reit any change
More informationChapter 10 Velocity, Acceleration, and Calculus
Chapter 10 Velocity, Acceleration, and Calculu The firt derivative of poition i velocity, and the econd derivative i acceleration. Thee derivative can be viewed in four way: phyically, numerically, ymbolically,
More informationPerformance of a BrowserBased JavaScript Bandwidth Test
Performance of a BrowerBaed JavaScript Bandwidth Tet David A. Cohen II May 7, 2013 CP SC 491/H495 Abtract An exiting browerbaed bandwidth tet written in JavaScript wa modified for the purpoe of further
More informationIs MarktoMarket Accounting Destabilizing? Analysis and Implications for Policy
Firt draft: 4/12/2008 I MarktoMarket Accounting Detabilizing? Analyi and Implication for Policy John Heaton 1, Deborah Luca 2 Robert McDonald 3 Prepared for the Carnegie Rocheter Conference on Public
More informationSolvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.
Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila
More informationStochastic House Appreciation and Optimal Subprime Lending
Stochatic Houe Appreciation and Optimal Subprime Lending Tomaz Pikorki Columbia Buine School tp5@mail.gb.columbia.edu Alexei Tchityi NYU Stern atchity@tern.nyu.edu February 8 Abtract Thi paper tudie an
More informationThe Nonlinear Pendulum
The Nonlinear Pendulum D.G. Simpon, Ph.D. Department of Phyical Science and Enineerin Prince Geore ommunity ollee December 31, 1 1 The Simple Plane Pendulum A imple plane pendulum conit, ideally, of a
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationGrowth and Sustainability of Managed Security Services Networks: An Economic Perspective
Growth and Sutainability of Managed Security Service etwork: An Economic Perpective Alok Gupta Dmitry Zhdanov Department of Information and Deciion Science Univerity of Minneota Minneapoli, M 55455 (agupta,
More informationProgress 8 and Attainment 8 measure in 2016, 2017, and 2018. Guide for maintained secondary schools, academies and free schools
Progre 8 and Attainment 8 meaure in 2016, 2017, and 2018 Guide for maintained econdary chool, academie and free chool September 2016 Content Table of figure 4 Summary 5 A ummary of Attainment 8 and Progre
More informationBUILTIN DUAL FREQUENCY ANTENNA WITH AN EMBEDDED CAMERA AND A VERTICAL GROUND PLANE
Progre In Electromagnetic Reearch Letter, Vol. 3, 51, 08 BUILTIN DUAL FREQUENCY ANTENNA WITH AN EMBEDDED CAMERA AND A VERTICAL GROUND PLANE S. H. ZainudDeen Faculty of Electronic Engineering Menoufia
More informationTransient turbulent flow in a pipe
Tranient turbulent flow in a pipe M. S. Ghidaoui A. A. Kolyhkin Rémi Vaillancourt CRM3176 January 25 Thi work wa upported in part by the Latvian Council of Science, project 4.1239, the Natural Science
More informationStochastic House Appreciation and Optimal Mortgage Lending
Stochatic Houe Appreciation and Optimal Mortgage Lending Tomaz Pikorki Columbia Buine School tp2252@columbia.edu Alexei Tchityi UC Berkeley Haa tchityi@haa.berkeley.edu December 28 Abtract We characterize
More informationDISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS
DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS Chritopher V. Kopek Department of Computer Science Wake Foret Univerity WintonSalem, NC, 2709 Email: kopekcv@gmail.com
More information1  Introduction to hypergraphs
1  Introduction to hypergraph Jacque Vertraëte jacque@ucd.edu 1 Introduction In thi coure you will learn broad combinatorial method for addreing ome of the main problem in extremal combinatoric and then
More informationQuadrilaterals. Learning Objectives. PreActivity
Section 3.4 PreActivity Preparation Quadrilateral Intereting geometric hape and pattern are all around u when we tart looking for them. Examine a row of fencing or the tiling deign at the wimming pool.
More informationINFORMATION Technology (IT) infrastructure management
IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 2, NO. 1, MAY 214 1 BuineDriven Longterm Capacity Planning for SaaS Application David Candeia, Ricardo Araújo Santo and Raquel Lope Abtract Capacity Planning
More informationResearch in Economics
Reearch in Economic 64 (2010) 137 145 Content lit available at ScienceDirect Reearch in Economic journal homepage: www.elevier.com/locate/rie Health inurance: Medical treatment v diability payment Geir
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science
aachuett Intitute of Technology Department of Electrical Engineering and Computer Science 6.685 Electric achinery Cla Note 10: Induction achine Control and Simulation c 2003 Jame L. Kirtley Jr. 1 Introduction
More informationUnobserved Heterogeneity and Risk in Wage Variance: Does Schooling Provide Earnings Insurance?
TI 011045/3 Tinbergen Intitute Dicuion Paper Unoberved Heterogeneity and Rik in Wage Variance: Doe Schooling Provide Earning Inurance? Jacopo Mazza Han van Ophem Joop Hartog * Univerity of Amterdam; *
More informationDesign of Compound Hyperchaotic System with Application in Secure Data Transmission Systems
Deign of Compound Hyperchaotic Sytem with Application in Secure Data Tranmiion Sytem D. Chantov Key Word. Lyapunov exponent; hyperchaotic ytem; chaotic ynchronization; chaotic witching. Abtract. In thi
More informationHow Enterprises Can Build Integrated Digital Marketing Experiences Using Drupal
How Enterprie Can Build Integrated Digital Marketing Experience Uing Drupal acquia.com 888.922.7842 1.781.238.8600 25 Corporate Drive, Burlington, MA 01803 How Enterprie Can Build Integrated Digital Marketing
More informationStochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations
36 Biophyical Journal Volume 8 December 200 36 336 Stochaticity in Trancriptional Regulation: Origin, Conequence, and Mathematical Repreentation Thoma B. Kepler* and Timothy C. Elton *Santa Fe Intitute,
More informationSimulation of Sensorless Speed Control of Induction Motor Using APFO Technique
International Journal of Computer and Electrical Engineering, Vol. 4, No. 4, Augut 2012 Simulation of Senorle Speed Control of Induction Motor Uing APFO Technique T. Raghu, J. Sriniva Rao, and S. Chandra
More informationUnusual Option Market Activity and the Terrorist Attacks of September 11, 2001*
Allen M. Potehman Univerity of Illinoi at UrbanaChampaign Unuual Option Market Activity and the Terrorit Attack of September 11, 2001* I. Introduction In the aftermath of the terrorit attack on the World
More informationProgress 8 measure in 2016, 2017, and 2018. Guide for maintained secondary schools, academies and free schools
Progre 8 meaure in 2016, 2017, and 2018 Guide for maintained econdary chool, academie and free chool July 2016 Content Table of figure 4 Summary 5 A ummary of Attainment 8 and Progre 8 5 Expiry or review
More informationMobility Improves Coverage of Sensor Networks
Mobility Improve Coverage of Senor Networ Benyuan Liu Dept. of Computer Science Univerity of MaachuettLowell Lowell, MA 1854 Peter Bra Dept. of Computer Science City College of New Yor New Yor, NY 131
More informationDUE to the small size and low cost of a sensor node, a
1992 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 14, NO. 10, OCTOBER 2015 A Networ Coding Baed Energy Efficient Data Bacup in SurvivabilityHeterogeneou Senor Networ Jie Tian, Tan Yan, and Guiling Wang
More informationA Duality Model of TCP and Queue Management Algorithms
A Duality Model of TCP and Queue Management Algorithm Steven H. Low CS and EE Department California Intitute of Technology Paadena, CA 95 low@caltech.edu May 4, Abtract We propoe a duality model of endtoend
More informationCHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY
Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Sidonia Otilia Cernea Mihaela Jaradat 2 Mohammad
More informationOUTPUT STREAM OF BINDING NEURON WITH DELAYED FEEDBACK
binding neuron, biological and medical cybernetic, interpike interval ditribution, complex ytem, cognition and ytem Alexander VIDYBIDA OUTPUT STREAM OF BINDING NEURON WITH DELAYED FEEDBACK A binding neuron
More informationSome remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s1366101505080 R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
More information