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On Some Elliptic Problems Involving Powers of the Laplace Operator On Some Elliptic Problems Involving Powers of the Laplace Operator by Alejandro Ortega Garc´ıa in partial fulfillment of the requirements for the degree of Doctor in Ingenier´ıaMatem´atica Universidad Carlos III de Madrid Advisor(s): Pablo Alvarez´ Caudevilla Eduardo Colorado Heras Tutor: Pablo Alvarez´ Caudevilla Legan´es,December 19, 2018 This Doctoral Thesis has been supported by a PIPF Grant from Departamento de Matem´aticasof Universidad Carlos III de Madrid. Additionally, the author has been par- tially supported by the Ministry of Economy and Competitiveness of Spain and FEDER under Research Project MTM2016-80618-P. Some rights reserved. This thesis is distributed under a Creative Commons Reconocimiento- NoComercial-SinObraDerivada 3:0 Espa~naLicense. \Sometimes science is more art than science, Morty. A lot of people don't get that." Agradecimientos Gracias, en primer lugar, a mis directores, Pablo y Eduardo, por haber hecho posible este tra- bajo. Sobra, por supuesto, decir lo que he aprendido bajo vuestra direcci´on.Sobra tambi´en enumerar las horas de pizarra y tel´efonoque me hab´eissufrido para aprenderlas. Todo eso va ´ımplicto en estas p´aginas.Estoy muy orgulloso de haber trabajado bajo vuestra direcci´on y os agradecer´esiempre la paciencia, la cercan´ıay amistad que me hab´eismostrado en todo momento. Quisiera tambi´endar las gracias al profesor Arturo de Pablo, contigo empez´omi andadura en esta universidad y mis primeros pasos en el mundo de los laplacianos fraccionarios. Gracias a los profesores Jos´eCarmona y Tommaso Leonori por la paciencia que han tenido conmigo y el trato que me han dado. En general, gracias al personal del Departamento de Matem´aticas de la UC3M por su amabilidad y disponibilidad siempre que lo he necesitado. Quiero tambi´enmostrar mi agradecimiento a los grandes trabajos sobre los que reposa esta tesis. Me sobrecoge la belleza y la profundidad de las Matem´aticasque he le´ıdoa lo largo de estos a~nos. Extra~noplacer el de comprender algo. Extra~naadicci´onla de buscar respuestas. Sin embargo, a´uncon toda su grandeza, las Matem´aticasno son nada sin las personas: Gracias Helena, sin ti no lo habr´ıaconseguido. Tuya es casi una mitad. A mi madre, Manuela, gracias por luchar porque lleguemos a ser lo que somos. A mi hermano, Jos´eManuel, por ense~narmeotros Teoremas, por la valent´ıay por las cosas que no caben en vocablos. A mi hermana, Mar´ıadel Carmen, por asombrarme con tu tenacidad y por ilusionarme sin saberlo. A ti, padre, que habitas los lugares m´asinesperados, qu´ete voy a contar que ya no hayamos hablado. A mis amigos, la familia elegida: Manuel, Sheriff, por serme incondicional y tan grande, matem´aticay personalmente. Carlos (& 5 from Memphis) por aquello de aguantar el tipo, sabes que soy tu fan n´umero1. Rula, por nuestras discusiones, por las cosas que me escuchas y -algunas de- las que me dices. Guille, por lo que sabes, por la actitud y lo imperceptible. Alvaro´ -algo conspiraba a nuestro favor- y Haza, por cerrar el c´ırculo,por las excursiones internacionales. A Rafael Granero, Rafa, por -ya sabes- ense~narmea jugar al ajedrez, y no tantas matem´aticas como cosas importantes. A C´esar, por nuestras conversaciones, por conocernos y quiz´aalgo por el cine y los libros. A Carlillos, porque el tiempo no pasa entre nosotros. A Jos´eRam´on,por confiar desde siempre en este proyecto, otro -de tantos- cabalgar hacia adentro. Decir nombres concretos siempre es peligroso. Gracias a todos. Cu´antos folios, cu´anta tinta. Saber es equivocarse muchas veces. Published and Submitted Content Chapter 3 is composed entirely by the results contained in the paper • E. Colorado, A. Ortega, The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions. Preprint arXiv:1805.10093v2, which I co-authored jointly with one of my advisors, Professor E. Colorado. This work was submitted to the scientific repository Arxiv.org. The material from this source included in this thesis is not singled out with typographic means and references. Contents Acknowledgements vii Nomenclature iii Summary of contents v Description of the results viii Part 1. Subcritical Problems with Mixed Dirichlet-Neumann Boundary data 1 Chapter 1. Regularity of solutions of a linear fractional elliptic problem with mixed Dirichlet-Neumann boundary conditions 3 1.1. Introduction 3 1.2. Functional setting and preliminaries 4 1.3. H¨olderRegularity 9 1.4. Moving the boundary conditions 28 Chapter 2. Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions 31 2.1. Introduction 31 2.2. Functional setting and preliminaries 33 2.3. Moving planes and monotonicity 34 2.4. A priori bounds in L1(Ω) 42 2.5. Minimal and mountain-pass solutions 44 2.5.1. Moving the boundary conditions 50 Part 2. Critical Problems with Mixed Dirichlet-Neumann Boundary data 53 Chapter 3. The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions 55 3.1. Introduction 55 3.2. Functional setting and definitions 57 3.3. Properties of the constants Se(ΣN ) and Se(ΣD) 59 3.4. Proof of main results 65 3.4.1. Proof of Theorem 3.1.1.(1)-(2) 65 3.4.2. Moving the boundary conditions. Proof of Theorem 3.1.1-(3) 71 3.5. A nonexistence result: Pohozaev-type identity 72 i ii CONTENTS Part 3. Critical Problems involving inverse operators and Dirichlet Boundary data 75 Chapter 4. Existence of positive solutions for a Brezis-Nirenberg{type problem involving an inverse operator 77 4.1. Introduction 77 4.2. Existence of positive solutions for the nonlocal problem (Pλ) 81 4.2.1. Concentration-Compactness for the nonlocal problem (Pλ) 84 4.3. Existence of positive solutions for the system (Sλ) 90 4.3.1. Concentration-Compactness for the system (Sλ) 93 4.4. Further Extensions 97 Chapter 5. Existence of positive solutions for a semilinear fractional elliptic equation involving an inverse fractional operator 103 5.1. Introduction 103 5.2. Subcritical exponent case 109 5.3. Concentration-Compactness at the critical exponent 118 5.3.1. Palais-Smale condition under the critical level 119 5.3.2. PS sequences under the critical level 125 Chapter 6. Homotopy Regularization for a High-Order parabolic equation 133 6.1. Introduction 133 6.2. Polyharmonic heat equation when n = 0 136 6.3. Preliminary estimates: Bernis{Friedman-type inequalities 138 6.4. Homotopy deformations 141 6.4.1. Branching of solutions from the Polyharmonic heat equation 144 Bibliography 149 Nomenclature Symbol Meaning n x = (x1; x2; :::; xN ) Element of R q 2 2 2 r = jxj = (x1 + x2 + ··· + xN ) Modulus of x h·; ·i Scalar product in RN @u @iu = = uxi Partial derivative of u respecto to xi @xi 2 2 @ u @iju = = uxixj Second partial derivative of u respect to xi and xj @xi@xj @u @u @u ru = ; ;:::; Gradient of u @x1 @x2 @xN @u @ν = hru; νi Outwards normal (to @Ω) derivative ∆u = div (ru) Laplacian of u (−∆)su Spectral Fractional Laplacian of u @w 1−2s @w @νs −κs limy!0+ y @y Γ(s) κs Normalizing constant equals to 21−2sΓ(1−s) 2N 2∗ = Critical fractional Sobolev exponent s N − 2s @Ω Boundary of Ω CΩ = Ω × (0; 1) Extension cylinder of Ω @LCΩ = @Ω × [0; 1) Lateral boundary of CΩ N BR(x0) Ball in R centered at x0 with radius R jAj Lebesgue measure of A ⊂ RN N jAj! Measure of A ⊂ R respect to the measure dµ = !dx χA Characteristic function of the set A k · kX Norm in the space X X0 Dual space of X n Difference of sets δx0 Dirac's delta centered at x0 δij Kronecker's delta a:e: Almost everewhere u+ = maxfu; 0g Positive part of the function u u− = max{−u; 0g Negative part of the function u C(Ω) or C0(Ω) Continuous functions in Ω C0(Ω) Continuous functions in Ω with compact support C0,γ(Ω) = Cγ(Ω) H¨oldercontinuous functions in Ω with exponent γ ju(x) − u(y)j γ jujγ = sup γ Seminorm in the space C (Ω) x;y2Ω jx − yj x6=y γ kukCγ (Ω) = kukC(Ω) + jujγ Norm in the space C (Ω) Symbol Meaning Ck(Ω) Functions of class k in Ω k C0 (Ω) Functions of class k in Ω with compact support C1(Ω) Functions infinitely differentiable in Ω 1 1 C0 (Ω) = D(Ω) Functions in C (Ω) with compact support 0 1 D Dual space of C0 (Ω), i.e. the space of distributions p R R p L (Ω); 1 ≤ p < 1 u :Ω 7! : u measurable; Ω juj dx L1(Ω) fu :Ω 7! R : u measurable and ju(x)j ≤ C a:e: in Ωg 1 1 H (Ω) Completeness of C0 (Ω) with the norm kφkH1(Ω) = kφkL2(Ω) + krφkL2(Ω) s n P 2 P s 2 o H (Ω) u = j aj'j 2 L (Ω) : kukHs(Ω) = j λjaj < 1 Hs (Ω) fu 2 Hs(Ω) : u = 0 on Σ ⊂ @Ωg ΣD D s C 1 C X0 ( Ω) Completeness of C0;L( Ω) with the norm 1=2 R 1−2s 2 kφk s C = κs y jrφj dxdy X0 ( Ω) CΩ X s (C ) Completeness of C1(Ω [ Σ ) × [0; 1) with the norm ΣD Ω 0 N 1=2 1−2s 2 s R kφkX (CΩ) = κs C y jrφj dxdy ΣD Ω 2s s N+2s N N 2π Γ(1−s)Γ( 2 )(Γ( 2 )) S(s; N) Sobolev constant equals to N−2s s Γ(s)Γ( 2 )(Γ(N)) Se(ΣD) Sobolev constant relative to ΣD ⊂ @Ω equals to 2 kukHs (Ω) ΣD inf 2 s kuk u2H (Ω) 2∗ ΣD L s (Ω) u6≡0 − 2s Se(ΣN ) Sobolev constant relative to ΣN ⊂ @Ω equals to 2 N S(s; N) Summary of contents This PhD Thesis is devoted to the study of some elliptic problems involving powers of the positive Laplace operator.
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