# Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.

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1 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 and Alexander Quaas (UTFSM, Chile) 1 / 28

2 Sections 1 Nonlocal Operators. 2 Supercritical problem. 3 Log-critical case. 2 / 28

3 1 Nonlocal Operators. 2 Supercritical problem. 3 Log-critical case. 3 / 28

4 Nonlocal operators For a nonnegative measurable function K : R N R and a function u : R N R, an elliptic nonlocal operator takes the form I(u, x) = [u(y) u(x)]k(x y)dy, R N each time the integral has a sense. The basic assumptions on K is the Lévy condition R N mín{1, y 2 }K(y)dy <. This implies growth at infinity and regularity requirements over u in order I(u) is well-defined. 4 / 28

5 Population dynamic interpretation. When K L 1 (R N ) = 1, the nonlocal heat equation u t = I(u), in R N (0, ) models the population dynamics driven by the nonlocal displacement of the individuals in the space. u(y)k(x y) K u(x)k(x y) y x y y y 5 / 28

6 The Fractional Laplacian The Fractional Laplacian of order σ for σ (0, 2) is given by ( ) σ/2 u(x) = C N,σ P.V. R N where C N,σ > 0 is a normalizing constant making ( ) σ/2 as σ 2. u(y) u(x) dy, x y N+σ It is an operator that differentiates σ-times. Comparison principle, regularizing effect (uniform ellipticity), variational structure, Fourier transform (hamonic analysis), infinitesimal generator of jump Lévy preocesses,... Lack of product rule, chain rule,... 6 / 28

7 1 Nonlocal Operators. 2 Supercritical problem. 3 Log-critical case. 7 / 28

8 The Supercritical Problem. For Ω R N bounded with smooth boundary, we are interested in the solvability of nonlocal Dirichlet problems with the form { λu + ( ) σ/2 u + Du p = f in Ω u = ϕ in Ω c (DP) where λ R, f C( Ω), ϕ C b (Ω c ). AIM: Solve (in the viscosity sense) the Dirichlet problem (DP) continuously in R N under the supercritical assumption. p > σ 8 / 28

9 Second-order case Let λ > 0. For the second-order problem { λu u + Du p = f in Ω u = ϕ on Ω (1) has a deep connection with stochastic optimal control problems. As it can be seen in Barles & Da Lio [JMPA 04]: Problem (1) is uniquely solvable by u C( Ω) satisfying the equation with generalized boundary conditions. If p 2 the boundary data is attained classically (u = ϕ on Ω). If p > 2, loss of the boundary condition phenomena may occur, that is, it may happen that u(x) ϕ(x) at some points x Ω. 9 / 28

10 Second-order case Capuzzo-Dolcetta, Leoni & Porretta [TAMS 10] prove that the superquadratic equation (1) can be continuously solvable (up to the boundary) provided Compatibility condition among f and ϕ: λ ínf{ϕ} ínf{f } Ω Ω Oscillation control of the boundary condition: There exists M > 0 small enough such that ϕ(x) ϕ(y) M x y p 2 p 1, for all x, y Ω. This last condition is in fact a compatibility condition with the C γ regularity of subsolutions to problem (1) (attaining or not the boundary condition). 10 / 28

11 The above assumptions allow to construct a function with the form w(x) Md(x) γ with M > 0 large enough to reach the constant subsolution, BUT not that large in order to have the viscosity inequality near the boundary. (easy) supersolution Ω solution w inf{f}/λ constant subsolution Key: Ellipticity. 11 / 28

12 Nonlocal problem For the nonlocal Dirichlet problem (DP) we have several similarities Barles, Chasseigne & Imbert [IUJM 08]: If p σ, problem (DP) has a unique viscosity solution u C(R N ) with u = ϕ on Ω c. Explicit examples of loss of the boundary condition when the gradient dominates. Barles & T. [SIMA 16]: Existence of a unique viscosity solution to problem (DP) with generalized boundary condition for any p > 0. Barles, Koike, Ley & T. [CVPDE 14]: When p > σ, interior C p σ p 1 estimates hold for problem (DP), and C p σ σ regularity estimates up to the boundary. Presence of jump discontinuities is fatal in the nonlocal setting. 12 / 28

13 Theorem (Dávila, Quaas & T.) Assume ϕ is supported on a set D R N compact with Ω D, and denote dy λ 0 (x) = C N,σ ínf (0, + ). x Ω x y N+σ D c Consider λ > λ 0. Assume that σ (1, 2), p (σ, σ/(2 σ)) and that ϕ, f satisfy (λ 0 + λ) ínf{ϕ} ínf {f } D Ω and ϕ(x) ϕ(y) M x y p σ p 1 for all x, y D \ Ω, for some M > 0 small enough. Then, there exists a unique viscosity solution u C p σ p 1 (R N ) to the problem (DP). 13 / 28

14 Following CD-L-P s paper: For each y Ω it is defined v y (x) = (ϕ(y) M x y γ )1 D (x) µmd γ +(x), where γ = (p σ)/(p 1). For suitable µ > 0, it is proven that v = ínf y Ω {v y } is a subsolution near the boundary (and attains the boundary data). Using the assumptions it is possible to extend the subsolution to the whole domain by gluing it with the constant subsolution. Once the barrier with Hölder profile γ is constructed, γ-hölder regularity up to the boundary for the solution follows standard arguments. 14 / 28

15 Remarks Role of the ellipticity: If σ > 1 and β < σ/2, then for some C > 0 and all x Ω close to the boundary ( ) σ/2 d β +(x) Cd β σ (x). d β +(x) x Ω The balance of powers precedure that leads to the result in the proof implies the condition p < σ/(2 σ). 15 / 28

16 Remarks General ϕ (not compactly supported): We artificially write ϕ = ϕ 1 + ϕ 2 with ϕ 1 = ϕ1 D and ϕ 2 = ϕ1 c D for some Ω D where ϕ is γ-hölder. We incorporate ( ) σ/2 ϕ 2 to the right-hand side f. The result is compatible with the second-order case when we make σ 2 (in an adequate way). Extension to fully nonlinear nonlocal problems: The upper bound for p is restricted by the ellipticity constants of the nonlocal operator. 16 / 28

17 1 Nonlocal Operators. 2 Supercritical problem. 3 Log-critical case. 17 / 28

18 Motivation Alarcón, García-Melián & Quaas [PRSEd 14]; and Kawohl & Kutev [AMSc 12] address the problem { λu u = g( Du ) + f in Ω u = 0 on Ω (2) where g C 1 [0, + ) R is increasing, coercive and g(0) = 0. They prove that (2) is continuously solvable if g satisfies the condition + 1 tdt = +, (3) g(t) allowing nonlinearities g with the asymptotic form g(t) = t 2 log α (t) for α < 1. Moreover, the solutions have bounded gradient at Ω, leading to classical solutions in several interesting cases. 18 / 28

19 Main idea Lemma Given M 1, M 2 > 0 and δ > 0, there exists L > 0 large enough such that the ode problem h (t) = g(h (t)) + M 1 t (0, δ) h(0) = 0, h(δ) = M 2, h (0) = L, and h 0 in (0, δ) has a (unique) solution h C[0, δ] C 2 (0, δ). In fact, denoting H(z) = H L,M1 (z) = then h has the explicit form h(t) = t 0 L z H 1 (y)dy. dy g(y) + M 1, 19 / 28

20 Main idea Then, it is possible to construct a supersolution to the problem with a function with the form v(x) = h(d(x)), x Ω. L h(d(x)) M 2 = f /λ δ Ω Key: Chain rule (to pass from the pde to the ode). 20 / 28

21 Fractional to ode Lemma (Dávila, Quaas & T.) Let σ (1, 2) and h : [0, + ] R a smooth, bounded function with h > 0, h < 0 and h monotone. Define v(x) = h(d + (x)). Then, for each x Ω close to the boundary we have ( ) σ/2 v(x) I 1 + I 2 + I 3 + I 4 where for some constants c, C, C 1, C 2, C 3 > 0 we denote I 1 := cd(x) 2 σ h (d(x)) Cd(x) 1 σ h(d(x)) I 2 := C 1 d(x) 1 σ h (d(x)) I 3 :=C 2 d σ (x)h(d(x)) I 4 := C 3 h. The constants C i 0 as σ / 28

22 Splitting of the integral Ω I 4 Ω c I 4 1 x d(x) I 1 I 3 I 2 The key estimate concerning I 1 is a consequence of the ellipticity in the gradient direction presented by Barles, Chasseigne & Imbert [JEMS 11]. 22 / 28

23 Ishii-Nakamura operator The term coming from I 2 creates serious problems to adapt the ode approach to a problem with the (full) fractional Laplacian. We present a log-critical result for the Ishii-Nakamura operator [CVPDE 10] defined as I(u, x) = C N,σ P.V. B(x,d(x)) u(y) u(x) dy, x y N+σ which have shown to be a valid approximation of the Laplacian (and p-laplacian) as σ 2 in bounded domains. 23 / 28

24 Ishii-Nakamura operator may be regarded as a degenerate nonlocal operator degenerating in the normal direction to the boundary. Ω d(x) x d(x) x 24 / 28

25 Theorem (Dávila, Quaas & T.) Given a superlinear nonlinearity g we define the function H(z) = + Assume g satisfies the condition + 1 z dy, for z > 0. g(y) th(t) 2 σ σ 1 dt = +. (4) g(t) Then, for all λ > 0 and f C( Ω) there exists a unique viscosity solution u C( Ω) to the problem { λu I(u) = g( Du ) + f in Ω (5) u = 0 on Ω which attainsthe boundary data continuously. 25 / 28

26 Observations The function g(t) = t σ satisfies (4) since a direct computation leads to + 1 dt t = +. A function with the form g(t) = t θ with θ > σ does not satisfy (4) since in that case the integral has the form + 1 dt t θ 1 σ 1 < +. A function with the form g(t) = t σ log α (t) with 0 < α < σ 1 satisfies (4). This is based on the fact that for such a g, the associated function H satisfies the estimate H(t) ct 1 σ log α (t) as t / 28

27 The splitting lemma suggest to solve the ode t 2 σ h (t) = g(h (t)) + M 1 t (0, δ) h(0) = 0, h(δ) = M 2, h (0) = L, and h 0 in (0, δ) with L large enough depending on M 1, M 2 and δ. In fact, this ode is integrable in (0, δ) and its solution h satisfies the requirements of the lemma. We use h(d(x)) as a barrier with M 2 = f /λ and M 1 = f + CM 2 where C > 0 is the constant coming from the splitting lemma. Finally, L >> 1 is fixed in terms of the data in order to control the extra term Cd 1 σ (x)h(d(x)) coming from the splitting lemma. 27 / 28

28 Thank you! 28 / 28

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