A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIAN. Keywords: shape optimization, eigenvalues, Dirichlet Laplacian


 Stephanie Barrett
 2 years ago
 Views:
Transcription
1 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA DORI BUCUR AD DARIO MAZZOLEI Abstract. In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy. Keywords: shape optimization, eigenvalues, Dirichlet Laplacian. Introduction and statement of the main results The results of this paper are motivated by spectral shape optimization problems for the eigenvalues of the Dirichlet Laplacian, e.g. min { λ k (Ω), Ω R, Ω = }, (.) where λ k denotes the k th eigenvalue of the Dirichlet Laplacian and the dimensional Lebesgue measure ( ). In order to prove existence of an optimal set Ω for problem (.), two different methods were proposed recently. On the one hand, in [7] it is proved a surgery result asserting that one can suitably modify an open set such that the first k eigenvalues of the Dirichlet Laplacian are not increasing, its measure remains constant and its diameter decrease below a certain threshold. This result together to the ButtazzoDal Maso existence theorem [] (which has a local character) gives a proof of global existence of solutions. By a different method, based on the so called shape subsolutions (see the definition in Section ), in [7] is proved the existence of solutions and moreover that all minimizers have finite diameter and finite perimeter. Recently, Van den Berg has studied in [4] a minimum problem with both a measure and a perimeter constraint: min { λ k (Ω), Ω R, Ω, Per(Ω) C }. (.) An existence result for this problem cannot be deduced from the results [7, 7]. The surgery method of [7] can hardly control the perimeter since the procedure generates new pieces of boundary which may have a large surface area. As well, in the presence of two simultaneous constraints, the notion of shape subsolution can not be used in a direct manner due to the lack of suitable Lagrange multipliers which can take into account both geometric constraints. The results of this paper are also intended to provide a tool allowing to prove existence of a solution for (.). In this paper we give a result which follows the main objectives of [7], but with the new requirement on the control of the perimeter. For this purpose, the surgery is done in a different
2 DORI BUCUR AD DARIO MAZZOLEI manner, using some of the key ideas of the shape subsolutions. Roughly speaking, we look at the local behavior of the torsion function and prove that if this function is small enough in some region, then one can cut out a piece of the domain controlling simultaneously the variation of the low part of the spectrum, of the measure and of the perimeter. Throughout the paper, by Ω we denote an open set of finite measure. For simplicity, and without restricting the generality, we shall assume that its measure is equal to. Here is our main result which, for clarity, is stated in a simplified way: Theorem.. For every K > 0, there exists D, C > 0 depending only on K and the dimension, such that for every open set Ω R with Ω = there exists an open set Ω satisfying () Ω =, diam (Ω) D and Per(Ω) min{per( Ω), C}, () if λ k ( Ω) K, then λ k (Ω) λ k ( Ω). The set Ω is essentially obtained by removing some parts of Ω and rescaling it to satisfy the measure constraint. In case the measure of Ω is not equal to, the constants D and C above depend also on Ω, following the rescaling rules of the eigenvalues, measure and perimeter We shall split the main result stated above in two distinct theorems, Thereoms 3.3 and 4.. The construction of Ω differs depending on which kind of control of the perimeter is desired. If the perimeter of Ω is infinite (or larger than C), it is convenient to use an optimization argument related to the shape subsolutions to directly construct the set Ω satisfying all the requirements above on eigenvalues, measure and diameter, but with a perimeter less than C (Theorem 3.3). If the perimeter of Ω is finite (for example smaller than C), we produce a different argument, by cutting in a suitable way the set Ω with hyperplanes, and removing some strips, decreasing in this way the perimeter (Thereom 4.) and of course satisfying all the requirements above on eigenvalues, measure and diameter. In this last case, the control of the perimeter is done through a De Giorgi type argument. We point out that the assertions of the two theorems are slightly stronger than the unified formulation stated in Theorem.. We note that the results of this paper hold true in exactly the same way if instead of open sets one works with quasiopen or measurable sets (see the precise definitions of the spectrum for this weaker settings in [0]). In general, if Ω is quasiopen or measurable, then the constructed set Ω is of the same type. In some situations in which the diameter of Ω is large, Ω could be chosen open and smooth.. The spectrum of the Dirichlet Laplacian and the torsion function Let Ω R be an open set of finite measure. Denoting by H0 (Ω) the usual Sobolev space, the eigenvalues of the Dirichlet Laplacian on Ω are defined by λ k (Ω) := min S k max u S k \{0} where the minimum ranges over all kdimensional subspaces S k of H 0 (Ω). Ω u dx Ω u dx, (.)
3 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA 3 The torsion function of Ω is the function denoted w Ω which minimizes the torsion energy E(Ω) := min Du dx udx, u H0 (Ω) R R and satisfies in a weak sense w Ω = in Ω, w Ω H 0 (Ω). ote that the torsion energy is negative if Ω and E(Ω) = w Ω dx < 0. R We recall (see for instance []) that if one extends the torsion function by zero on R \ Ω, then it satisfies w Ω in the sense of distributions in R. A fundamental property of the torsion function is the Saint Venant inequality, which states that among all (open) sets of equal volume, the ball maximizes the L norm of the torsion function. This leads to the following inequality Ω w Ω dx Ω + ω / ( + ), (.) where ω is the volume of the ball of radius in R. A similar inequality between the L norms was proved by Talenti in [8] and leads to : w Ω L ( Ω ) ω. (.3) We recall the following bound on the ratio between eigenvalues of the Dirichlet Laplacian, which can be found in []. For all k there exists a constant M k, depending only on k and the dimension, such that λ k(ω) λ (Ω) M k. (.4) Another fundamental inequality, proved in [3] (see also [5]), relates the L norm of the torsion function with the first eigenvalue and reads λ (Ω) w Ω log. (.5) λ (Ω) We also recall the following inequality due to Berezin, Li and Yau (see [6]), which asserts that for some constant C depending only on the dimensions of the space, we have k ( k ) λ k (Ω) C. Ω The way we shall use this inequality is the following: if one fixes K > 0, then the number of ( ) eigenvalues of Ω below K, is at most of K Ω. C The γdistance between two open sets with finite measure Ω, Ω is defined by: d γ (Ω, Ω ) := w Ω w Ω dx. R
4 4 DORI BUCUR AD DARIO MAZZOLEI For sets satisfying Ω Ω, the following inequality was proved in [7] : for every k λ k (Ω ) k e /4π λ k (Ω ) / d γ (Ω, Ω ), (.6) λ k (Ω ) and we notice that there is a strong relation between the γdistance and the torsion energy: d γ (Ω, Ω ) = (E(Ω ) E(Ω )). Let c > 0. It is said that Ω R is a shape subsolution for the energy if for all Ω Ω E(Ω) + c Ω E( Ω) + c Ω. It is proved in [7] that, if Ω is a shape subsolution for the energy, then it is bounded (with controlled diameter) and has finite perimeter. We conclude this Section with a result relating a pointwise value of the torsion function to its integral on some neighborhood. Lemma.. Let Ω R be an open set and w = w Ω be its torsion function. For every θ > 0, there exists δ 0 > 0 depending only on, θ such that if w(x) θ for some x R, then wdx θω δ, δ (0, δ 0 ). B δ (x) Proof. Since for every x 0 R the function x w(x) + x x 0 that, for all δ > 0 θ w(x 0 ) B δ B δ (x 0 ) is subharmonic in R, we have (w(x) + x x 0 ) dx = δ wdx + B δ B δ (x 0 ) ( + ). For some δ 0 sufficiently small (e.g equal to θ( + )), we have 0 < δ δ 0 wdx θω δ. B δ (x) 3. Control of the spectrum by subsolutions Before stating our first result, we outline the main ideas. Let Ω R be a given open set of finite measure. Assume that for some set Ω Ω and for some constant c > 0 we have E(Ω) + c Ω E( Ω) + c Ω. (3.) ) Ω are Then, we shall observe that a certain number of low eigenvalues of the rescaled set not larger than the corresponding eigenvalues on Ω, provided that c is small enough. Smaller is the constant c, more eigenvalues satisfy this property. Indeed, from (.6), we get ( Ω Ω λ k (Ω) λ k ( Ω) 4k e /4π λ k (Ω)λ k ( Ω) (+)/ [E(Ω) E( Ω)]. (3.) Setting K Ω, Ω = 4k e /4π λ k (Ω)λ k ( Ω) (+)/, using inequality (3.) we get λ k (Ω) λ k ( Ω) ck Ω, Ω( Ω Ω ) ck Ω, Ω Ω ( Ω Ω ). (3.3)
5 we get so A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA 5 Then, for every Λ such that ck Ω, Ω Ω λ k (Ω) λ k ( Ω) Λ( Ω Ω ), λ k (Ω) + Λ Ω λk ( Ω) + Λ Ω. Λ (3.4) If c is small enough so that we can choose Λ satisfying λ k ( Ω) = Λ Ω, we get λ k (Ω) Ω λk ( Ω) Ω. The construction above can be carried out provided that one has control on an upper bound of K Ω, Ω in (3.4). We shall prove that this is the case, if c is small enough. Lemma 3.. Let k, K > 0 and Ω R be an open set of unit measure, satisfying λ k ( Ω) K. There exist two constants c, β > 0 depending only on K and such that for all Ω Ω satisfying we have and Ω β Ω. Proof. We divide the proof in several steps. E(Ω) + c Ω E( Ω) + c Ω (3.5) λ i (Ω) Ω / λ i ( Ω) Ω /, i =,..., k (3.6) Step. The constant c can be chosen such that E( Ω) + c Ω is negative. In order to find the right information on c, we start by proving the following inequality: w Ωdx C(), (3.7) Ω (λ ( Ω)) + with C() := () + ω (+). We first note that (w Ω λ ( Ω) )+ is the torsion function of the set {w Ω > } and that λ ( Ω) w Ω λ ( Ω) /λ ( Ω), thanks to (.5). As a consequence of the Talenti inequality (.3), the measure of the set {w Ω > } is controlled from below by λ λ ( Ω) ( Ω), precisely we have w Ωdx Ω { } w Ω> λ ( Ω) ( Then it is clear that we have w Ω λ ( Ω) E( Ω) + c Ω ) dx + { } w Ω> λ ( Ω) C() (λ ( Ω)) + The right hand side above is negative, as soon as we choose since λ ( Ω) K. c C() Ω (K) + λ ( Ω) dx C(). (λ ( Ω)) + + c Ω., (3.8)
6 6 DORI BUCUR AD DARIO MAZZOLEI Step. The constant c can be chosen such that for every Ω satisfying (3.5), we have w Ω w Ω. Indeed, denote h := w Ω and assume that w Ω < w Ω. Then 0 c Ω c Ω + w Ω dx w Ωdx = c Ω + w Ω dx min { w Ω, h/ } dx ( w Ω h/ ) dx c Ω {w Ω>h/} ( w Ω h/ ) dx. {w Ω>h/} Thanks to the fact that (w Ω h/) + = w {w using the same argument as in Step, we Ω>h/} have that C()h + (w Ω h/) + c Ω. {w Ω>h/} Consequently, if c C()K + then w Ω w Ω, since by inequality (.5) w Ω λ ( Ω) K. We note that, using the inequalities (.4) and (.5) together with the fact that w Ω w Ω /, one can easily deduce that for every i the corresponding eigenvalues on Ω and Ω are comparable λ i ( Ω) λ i (Ω) (8 + 6 log )M i λ i ( Ω). (3.9) Step 3. Proof of inequality (3.6). Choosing c satisfying Steps and, and λ (B) c, (3.0) M k k (8 + 6 log )e /4π k K + where B is the ball of volume equal to, from the FaberKrahn inequality we have or this precisely gives, in view of (3.)(3.3), cm k k (8 + 6 log )e /4π k K + λ ( Ω), (3.) Ω / λ i (Ω) + Λ Ω / λ i ( Ω) + Λ Ω /. (3.) Thanks to Step, c is such that Λ λ ( Ω) Ω /. Consequently, (3.) also holds for all values larger than λ ( Ω), thus this inequality holds for all i =,..., k with constant λ i( Ω). Ω / Ω / λ With the choice of i ( Ω) Ω /, using the arithmetic geometric inequality, we note that λ i (Ω) Ω / λ i ( Ω) Ω /, i =,..., k. Step 4. In order to control the diameter of the rescaled set, we prove the existence of β > 0 such that Ω β Ω. Indeed, we have the chain of inequalities (the last one being a consequence of (.3)) K λ ( Ω) w Ω w Ω which gives the estimate for β, depending only on K,. ( Ω ) ω,
7 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA 7 Remark 3.. Inequality (3.9) in Step could also be obtained in a different way, as a consequence of inequality (.6) by choosing c small enough. Indeed, if c is such that k e /4π λ k ( Ω) / d γ (Ω, Ω) λ k ( Ω), (3.3) then λ k ( Ω) λ k (Ω) λ k ( Ω). Since by the hypothesis of Lemma 3. we have inequality (3.3) holds as soon as E(Ω) E( Ω) c, c 8k e /4π λ /+ k ( Ω). ow, we are in position to prove the first result. Below, the diameter of a disconnected set is referred as the sum of the diameters of each connected component. Theorem 3.3. For every K > 0, there exists D, C > 0 depending only on K and the dimension such that for every open set Ω R with Ω = there exist an open set Ω with diam (Ω) D, Ω =, Per(Ω) C and if λ k ( Ω) K then λ k (Ω) λ k ( Ω). Proof. From the BerezinLiYau inequality, the maximal index k 0 for which it is possible that λ k0 ( Ω) K is lower than a constant depending only on K and. Let us consider the minimum problem min {E(Ω) + c Ω }, Ω Ω with c the constant given by Lemma 3. and k = k 0. This problem has at least one solution, denoted Ω, which is an open set (see for instance [4]) and it is also a shape subsolution of the energy. The results from [7] give that diam (Ω ) D(c) and that Per(Ω ) C(c) and we remind that c depends only on K and the dimension. Moreover, using Step 4 of Lemma 3., we have that the set Ω := Ω / Ω has still diameter and perimeter bounded by constants depending only on K,, thanks to the fact that Ω β Ω. Moreover we have that i =,..., k, λ i (Ω) λ i ( Ω), since λ k ( Ω) K. 4. Control of the perimeter In order to give precise statements, we introduce a suitable notion of diameter in a prescribed direction. In the coordinate direction e R we set diam e (Ω) := H ( t R : H (Ω {x = t}) > 0 ). Theorem 4.. For every K, P > 0, there exist D > 0 depending only on K, P and the dimension, such that for every open set Ω R with Ω =, Per( Ω) P, there exists an open set Ω of unit measure with diam e (Ω) D, Per(Ω) Per( Ω) such that if λ k ( Ω) K then λ k (Ω) λ k ( Ω).
8 8 DORI BUCUR AD DARIO MAZZOLEI Re by For every x R and r > 0, we define the strip centered in x of width r orthogonal to S r (x ) := [ r + x, r + x ] R. Its topological boundary is S r := { r + x, r + x } R. If x = 0, we simply denote S r instead of S r (0). The main idea of the following lemma is inspired from [] and was also used in [], [7], and [8] under different settings. We point out that here we do not use optimality, but only an inequality between two fixed domains. Lemma 4.. For all c > 0, there exist C 0, r 0 > 0, with C 0 r 0 min { c, K } such that if for some r r 0 the function w is not identically zero in S r and E( Ω) + c Ω E( Ω \ S r ) + c Ω \ S r, (4.) then max S r w Ω C 0 r. (4.) Proof. Below, we denote w := w Ω and ε := max Sr w and introduce the function η : R R + : η = 0 in S r, η = ε in R \ S r, η = in S r \ S r, (4.3) η = 0 on S r, η = ε on S r. Since the function min {w, η} := w η belongs to H0 ( Ω \ S r ) we get E( Ω \ S r ) D(w η) dx w ηdx. Hypothesis (4.) gives Dw dx wdx + c S r Ω D(w η) dx w ηdx. Since ε C 0 r 0 c we get w η = w in Ω \ S r. Denoting the outer unit normal to a set by ν, Dw dx + c S r S r Ω Dw dx wdx + c S r Ω S r S r ( D(w η) Dw )dx (w η w)dx S r \S r S r \S r = ( Dη Dw )dx (w η) + dx S r \S r {w>η} S r \S r Dη D(w η)dx (w η) + dx S r \S r {w>η} S r \S r η = S r ν (w η)+ dx = η (r) w dh. S r
9 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA 9 The following trace inequality holds: S r w dh C() ( ) wdx + Dw dx. r S r S r By using hypothesis (4.) and the CauchySchwarz inequality on the gradient term, we get to ( w dh C() (C 0 + S r ) S r Ω + ) Dw dx. If S r Dw dx + S r Ω = 0 then w = 0 in the strip S r. Otherwise S r Dw dx + S r Ω > 0 and from the previous inequality we get { min, c } η (r) C()(C 0 + ). Since η (r) = C 0 r/, choosing C 0 and r 0 small enough we get a contradiction. We notice that the choice of these constants depends only on and on c. S r Ω The following corollary can be proved in the very same way as Lemma 4.. Corollary 4.3. For all c > 0 there exist C 0, r 0 > 0 with C 0 r 0 min { c, K } such that if for some r r 0 and x,..., x n R such that S r (x i ) S r (x j ) = for all i j, it holds max { w Ω(x) : x i S r (x i ) } C 0 r 0, then we have that E( Ω \ i S r (x i )) + c Ω \ i S r (x i ) E( Ω) + c Ω. (4.4) Here we outline the main idea for proving Theorem 4.. Let c be as in Lemma 3. and r 0, C 0 be the constants from Lemma 4., for that particular choice of c. We shall remove a finite number of strips S r (x i ) from the region where w Ω(x) C 0 r 0 thus, following inequality (4.4) and Lemma 3., we can control the eigenvalues after rescaling. The control of the perimeter, will be done by a suitable choice of the position of the strips. Contrary to the construction in [7], the new perimeter introduced by sectioning with hyperplanes does not depend on the H measure of the boundary of the sections. Let l 0 > 0 and n. The value of l 0 will be precised below, in Lemma 4.4. Assume x i R and L i > l 0, i =,..., n are such that S Li (x i ) S L j (x j ) = if i j. For every t [0, l 0 ] we define: S(t) := n i=s Li t(x i ). For an open set of unit measure Ω, we denote m(t) := S(t) Ω the mass of the union of strips in Ω and σ(t) := n H ( Ω {L i t, L i + t} R ), i= the new perimeter introduced by the sections with the hyperplanes and n p(t) = Per( Ω (L i t, L i + t) R ) σ(t), i=
10 0 DORI BUCUR AD DARIO MAZZOLEI the perimeter of Ω inside the strips. We denote the rescaled set, Ω(t) := ( m(t)) / ( Ω \ S(t)). Lemma 4.4. Given P > 0 and an open set Ω of unit measure, with Per( Ω) P, there exist two constants l 0 and m, depending only on P and the dimension, such that if m(l 0 ) m then there exists t [0, l 0 ] such that Per(Ω(t)) Per( Ω). Proof. First of all, we notice that, by definition, t m(t) is a nonincreasing function and for a.e. t (0, l 0 ), we have that σ(t) = m (t). If for every t [0, l 0 ] we would have Per(Ω(t)) > Per( Ω), we get: Per( Ω) p(t) + σ(t) Per( Ω)( m(t)). There exists a constant m (depending only on P, ), such that if m(t) m, then ( m(t)) m(t) P m(t) Per( Ω). Putting the above inequalities together and using the isoperimetric inequality for the set S(t), Per( Ω) + σ(t) Per( Ω) Since ω / m(t) > 0, we obtain: + p(t) + σ(t) Per( Ω) m(t) + ω / m(t). By integrating on [0, l 0 ] we get m (t) (ω / Since m(0) = m and m(l 0 ) 0, choosing l 0 > )m(t). 4 m / (0) m / (l 0 ) (ω / ) l ω / m/ we get a contradiction. Remark 4.5. If we denote by A a subset of Ω with max A w Ω C 0 r 0 then, having in mind (.5), if m is small enough (depending only on C 0 and r 0 ) we get λ (A)( m) / C 0 r 0 K. Remark 4.6. Thanks to the choice of C 0, r 0 made in Lemma 4., we deduce that if A Ω is such that max A w Ω C 0 r 0, then E(A) + c A 0. Indeed, using the monotonicity of the torsion function: E(A) + c A = w A dx + c A since C 0 r 0 c from the hypotheses of Lemma 4.. A w Ωdx + c A C 0r 0 A We are now in position to prove the main result of this section. + c A 0,
11 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA Proof of Theorem 4.. We fix the constant c such that Lemma 3. is satisfied, we get C 0, r 0 from Lemma 4. and we fix a constant m that works both for Lemma 4.4 and for Remark 4.5. For simplicity we rename w = w Ω. The region where w(x) C 0 r 0 is contained in a finite union of strips with width 4r 0. Indeed, we define { X 0 := x R : max w C 0r 0 S r0 (x ) }, X := { S r0 (t) : t X 0 }. From Lemma. and the Saint Venant inequality (.) the set X is contained in the union of at most n = n(r 0, ) of disjoint strips (each of width at least 4r 0 ). Let us call X the projection of X on Re. The set R \ X is a finite union of disjoint segments and of the infinite intervals at ±, say [ n ] R \ X = (, b 0 ) (a i, b i ) (a n+, ). i= If a segment (a i, b i ) has a length less than or equal to 8r 0 + l 0, we shall ignore it in our further construction and just add the corresponding strip to the set X and renumber the index i if necessary. The total length of those such segments is at most n(8r 0 + l 0 ). Therefore, we shall assume in the sequel that all segments (a i, b i ) have a length greater than 8r 0 + l 0. We denote a i = a i + (4r 0 + l 0 ), b i = b i (4r 0 + l 0 ) and [ n+ ] [ n ] Y = (a i, a i ) (b i, b i ). i= In order to highlight the main idea, let us assume in a first instance that i=0 ( Y R ) Ω m. (4.5) If we are in this situation, we perform a simultaneous cut as in Lemma 4., removing the following union of strips: S t := S r0 (b 0 r 0 t) S r0 (a i + r 0 + t) S r0 (b i r 0 t) S r0 (a n+ + r 0 + t), for every t [0, l 0 ]. Following the assumption (4.5) and Lemma 4.4, there exists a value t such that the perimeter of the rescaled set Ω \ S t ( Ω \ S t ) is at most Per( Ω). Moreover, from the choice of c and Lemma 4., all the eigenvalues less than K of the rescaled set are not greater than the ones on Ω. In order to handle the diameter of the rescaled set, we replace all the connected components having a projection on Re disjoint from X by one ball, such that the volume remains unchanged. In this way, the perimeter does not increase, while the low part of the spectrum (below K) can only decrease, since the first eigenvalue of every such a connected component is not smaller than /(C 0 r 0 ) K (see Remark 4.5). It is clear that the new set satisfies the diameter bound: ( ) diam e (Ω) diam e ( Ω)( m) / H (X) + n(8r 0 + l 0 ) + r 0 (n + ) + ω.
12 DORI BUCUR AD DARIO MAZZOLEI If assumption (4.5) does not hold, we can not apply directly Lemma 4.4. Let p depending only on P and the dimension, be such that p m < p. If a i + p(4r 0 + l 0 ) > b i p(4r 0 + l 0 ), we ignore this strip and add it to X, renumbering the index i if necessary. There exists s [0, p ] such that replacing simultaneously all a i with a i +s(4r 0 +l 0 ) and b i with b i s(4r 0 +l 0 ) the assumption (4.5) is satisfied and so we finish the proof, adding at worst 4np(4r 0 + l 0 ) to the diameter. Remark 4.7. Since the choice of the direction e was arbitrary, we can repeat all the process of the proof of Theorem 4. for all the coordinate direction, finding a set which has diameter bounded in all directions, unit measure, better eigenvalues than Ω up to level k and perimeter lower than Ω. Acknowledgments. This work was initiated during the first author stay at the Isaac ewton Institute for Mathematical Sciences Cambridge in the programme Free boundary problems and related topics, as an answer to a question raised by Michiel Van den Berg. The first author also acknowledges the support of AR BS Optiform. The work of the second author has been supported by the ERC Starting Grant n AnOptSetCon. References [] H. W. Alt, L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 35 (98), [] M.S. Ashbaugh, The universal eigenvalue bounds of PaynePölyaWeinberger, HileProtter, and H C Yang, Proc. Indian Acad. Sci. (Math. Sci.) () (00), [3] M. van den Berg, Estimates for the torsion function and Sobolev constants, Potential Anal., 36 (4) (0) [4] M. van den Berg, On the minimization of Dirichlet eigenvalues, preprint available at arxiv: [5] M. van den Berg, D. Bucur, On the torsion function with Robin or Dirichlet boundary conditions, Journal of Functional Analysis (03), [6] T. Briançon, M. Hayouni, M. Pierre, Lipschitz continuity of state functions in some optimal shaping, Calc. Var. PDE, 3 () (005), 3 3. [7] D. Bucur, Minimization of the kth eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal. 06 (0) [8] D. Bucur, G. Buttazzo, B. Velichkov, Spectral optimization for potentials and measures, SIAM Journal of Mathematical Analysis, to appear (04). [9] D. Bucur, G. Buttazzo, Variational methods in shape optimization problems, Progress in nonlinear differential equations and their applications, Birkhäuser Verlag, Boston (005). [0] D. Bucur, D. Mazzoleni, A. Pratelli, B. Velichkov, Lipschitz regularity of the eigenfunctions on optimal domains, to appear on Arch. Ration. Mech. Anal. (04). [] G. Buttazzo, G. Dal Maso, An existence result for a class of shape optimization problems, Arch. Ration. Mech. Anal. (993), [] G. De Philippis, B. Velichkov, Existence and regularity of minimizers for some spectral functionals with perimeter constraint. Appl. Math. Optim. 69 (04), no., [3] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel (006).
13 A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA 3 [4] M. Hayouni, Lipschitz continuity of a state function in a shape optimization problem, J. Convex Anal., 6 (999) (), [5] A. Henrot, M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48, Springer (005). [6] P. Li, S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (983), no. 3, [7] D. Mazzoleni, A. Pratelli, Existence of minimizers for spectral problems, J. Math. Pures Appl., 00 (03), [8] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola orm. Sup. Pisa Cl. Sci., 3 (976) (4), Institut Universitaire de France and Laboratoire de Mathématiques (LAMA) UMR CRS 57, Université de Savoie, Campus Scientifique, LeBourgetDuLac  FRACE address: Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata,, 700 Pavia  ITALY address:
A 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 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 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 informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationNOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σalgebras 7 1.1 σalgebras............................... 7 1.2 Generated σalgebras.........................
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More informationExistence de solutions à support compact pour un problème elliptique quasilinéaire
Existence de solutions à support compact pour un problème elliptique quasilinéaire Jacques Giacomoni, Habib Mâagli, Paul Sauvy Université de Pau et des Pays de l Adour, L.M.A.P. Faculté de sciences de
More informationDiscussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine GenonCatalot To cite this version: Fabienne
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More information) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.
Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier
More informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More information1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification
More informationMathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More informationCHARLES D. HORVATH AND MARC LASSONDE. (Communicated by Peter Li)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 4, April 1997, Pages 1209 1214 S 00029939(97)036228 INTERSECTION OF SETS WITH nconnected UNIONS CHARLES D. HORVATH AND MARC LASSONDE
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationNotes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380  Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationIf the sets {A i } are pairwise disjoint, then µ( i=1 A i) =
Note: A standing homework assignment for students in MAT1501 is: let me know about any mistakes, misprints, ambiguities etc that you find in these notes. 1. measures and measurable sets If X is a set,
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationExercise 1. Let E be a given set. Prove that the following statements are equivalent.
Real Variables, Fall 2014 Problem set 1 Solution suggestions Exercise 1. Let E be a given set. Prove that the following statements are equivalent. (i) E is measurable. (ii) Given ε > 0, there exists an
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal RodríguezBernal Alejandro VidalLópez Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationSolutions to Tutorial 2 (Week 3)
THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 2 (Week 3) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2016 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/
More informationIn memory of Lars Hörmander
ON HÖRMANDER S SOLUTION OF THE EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the equation in the plane with a weight which permits growth near infinity carries
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationm = 4 m = 5 x x x t = 0 u = x 2 + 2t t < 0 t > 1 t = 0 u = x 3 + 6tx t > 1 t < 0
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 21, Number 2, Spring 1991 NODAL PROPERTIES OF SOLUTIONS OF PARABOLIC EQUATIONS SIGURD ANGENENT * 1. Introduction. In this note we review the known facts about
More informationRESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS
RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationOn the Union of Arithmetic Progressions
On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationStructure of Measurable Sets
Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationLECTURE NOTES: FINITE ELEMENT METHOD
LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite
More informationREAL ANALYSIS I HOMEWORK 2
REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other
More information5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationConductance, the Normalized Laplacian, and Cheeger s Inequality
Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 21, 2015 Disclaimer These notes are not necessarily an accurate representation
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationVariational approach to restore pointlike and curvelike singularities in imaging
Variational approach to restore pointlike and curvelike singularities in imaging Daniele Graziani joint work with Gilles Aubert and Laure BlancFéraud Roma 12/06/2012 Daniele Graziani (Roma) 12/06/2012
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).
Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationThe Geometry of Graphs
The Geometry of Graphs Paul Horn Department of Mathematics University of Denver May 21, 2016 Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Graphs Ultimately, I want
More informationDISTRIBUTIONS AND FOURIER TRANSFORM
DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationOn the representability of the biuniform matroid
On the representability of the biuniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every biuniform matroid is representable over all sufficiently large
More informationWissenschaftliche Artikel (erschienen bzw. angenommen)
Schriftenverzeichnis I. Monographie [1] Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, BerlinHeidelberg New York, 2003.
More informationNonlinear Fourier series and applications to PDE
CNRS/Cergy CIRM Sept 26, 2013 I. Introduction In this talk, we shall describe a new method to analyze Fourier series. The motivation comes from solving nonlinear PDE s. These PDE s are evolution equations,
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
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 information1 Introduction, Notation and Statement of the main results
XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 2428 septiembre 2007 (pp. 1 6) Skewproduct maps with base having closed set of periodic points. Juan
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationFinite speed of propagation in porous media. by mass transportation methods
Finite speed of propagation in porous media by mass transportation methods José Antonio Carrillo a, Maria Pia Gualdani b, Giuseppe Toscani c a Departament de Matemàtiques  ICREA, Universitat Autònoma
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 informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMATH41112/61112 Ergodic Theory Lecture Measure spaces
8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationSur un Problème de Monge de Transport Optimal de Masse
Sur un Problème de Monge de Transport Optimal de Masse LAMFA CNRSUMR 6140 Université de Picardie Jules Verne, 80000 Amiens Limoges, 25 Mars 2010 MODE 2010, Limoges Monge problem t =t# X x Y y=t(x) Ω f
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationA class of infinite dimensional stochastic processes
A class of infinite dimensional stochastic processes John Karlsson Linköping University CRM Barcelona, July 7, 214 Joint work with JörgUwe Löbus John Karlsson (Linköping University) Infinite dimensional
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More information