<p>The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of L\'evy processes.</p> <p>Firstly, we prove inequalities on first exit time distributions in the case that the L\'evy process is an $\alpha$-stable symmetric process $A_t$ on $\R^d$, $\alpha\in(0,2]$. Given $A_t$ and a bounded domain $D\subset\R^d$, we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B\subset\R^d$ and satisfying the $\varepsilon$-cone property converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from  $D_m$  converges to the distribution of the exit time of Brownian motion from $D'$. The second set of results in this thesis extends the theorems from \cite{BanMen} by proving generalized isoperimetric inequalities on expectations of L\'evy processes in the case of Steiner symmetrization.% using the Brascamp-Lieb-Luttinger inequalities used above. </p> <p>These results will then be used to establish inequalities involving distributions of first exit times of $\alpha$-stable symmetric processes $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.</p>

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space. / Målet med detta arbete är att vara en introduktion på kandidatnivå till ämnesfältet potentialteori, genom att studera ett viktigt matematiskt problem inom potentialteori, Dirichletproblemet. Genom att undersöka den historiska utvecklingen av olika lösningsmetoder för problemet i mer och mer allmänna sammanhang, i kombination med de matematiska koncepten som utvecklades för att användas i dessa lösningsmetoder, ges en översikt av olika grundläggande tekniker inom potentialteori, samt en sammanfattaning av de olika matematiska resultaten som har att göra med Dirichletproblemet i det Euklidiska rummet.

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-17T20:37:08Z No. of bitstreams: 1 Dissertação-Andrea M da Mota.pdf: 827395 bytes, checksum: 3b513795b0e557b49dc4814527d37611 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-19T14:12:03Z (GMT) No. of bitstreams: 1 Dissertação-Andrea M da Mota.pdf: 827395 bytes, checksum: 3b513795b0e557b49dc4814527d37611 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-19T14:14:29Z (GMT) No. of bitstreams: 1 Dissertação-Andrea M da Mota.pdf: 827395 bytes, checksum: 3b513795b0e557b49dc4814527d37611 (MD5) / Made available in DSpace on 2015-06-19T14:14:29Z (GMT). No. of bitstreams: 1 Dissertação-Andrea M da Mota.pdf: 827395 bytes, checksum: 3b513795b0e557b49dc4814527d37611 (MD5) Previous issue date: 2014-09-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The objective of this notes is to prove in detail a theorem, due to Ernst Heintze and Hermann Karcher, establishing an upper bound for the volume of compact domains in a connected closed hypersurface immersed in Euclidean space E. As application we will give an alternative proof of the Alexandrov’s theorem, which states that the Euclidean spheres are the only embedded closed hypersurfaces of constant mean curvature in E. / O objetivo deste trabalho é demonstrar em detalhes um teorema devido a Ernst Heintze e Hermann Karcher que estabelece uma cota superior para o volume de domínios compactos em uma hipersuperfície conexa, fechada e mergulhada no espaço euclidiano E. Como aplicação será dada uma prova alternativa do Teorema de Alexandrov, que caracteriza as esferas euclidianas como as únicas hipersuperfícies conexas, fechadas e mergulhadas de curvatura média constante em E.

The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis. An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem. An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived. The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control. Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.

The thesis consists of four research papers. The first three deal with the Choquet theory of function spaces. In Chapter 1, a theory on products and projective limits of function spaces is developed. It is shown that the product of simplicial spaces is a simplicial space. The stability of the space of maximal measures under continuous affine mappings is studied in Chapter 2. The third chapter employs results from the previous chapters to construct an example of a function space where the abstract Dirichlet problem is not solvable for any class of Baire-n functions with $n\in N$. It is shown that such an example cannot be constructed via the space of harmonic functions. In the final chapter, the recently introduced class of sequentially Right Banach spaces is being investigated. Connections to other isomorphic properties of Banach spaces are established and several characterizations are given.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-08T20:06:07Z No. of bitstreams: 1 JOSÉ DE BRITO SILVA - DISSERTAÇÃO PPGMAT 2013..pdf: 535262 bytes, checksum: eb7f0d4f7e69b8a4b86d3e1dc0f16739 (MD5) / Made available in DSpace on 2018-08-08T20:06:07Z (GMT). No. of bitstreams: 1 JOSÉ DE BRITO SILVA - DISSERTAÇÃO PPGMAT 2013..pdf: 535262 bytes, checksum: eb7f0d4f7e69b8a4b86d3e1dc0f16739 (MD5) Previous issue date: 2013-10 / Capes / Neste trabalho discutiremos a existência de soluções fracas positivas para um sistema do (p, q)-Laplaciano com mudança de sinal nas funções de peso, com domínio limitado e fronteira suave. Para garantir a existência de soluções fracas positivas primeiramente asseguraremos a solução positiva de um problema calásico que é o problema de autovalor do p-laplaciano, e do problema "linear"do p-laplaciano com condição zero de Dirichlet. Feito isto usaremos a existência destas soluções para assegurar que o problema em questão admite solução fraca positiva, via o método das sub-super-soluções / In this work we discuss the existence of weak positive solutions for a system (p, q)- Laplacian with change of sign in the weight functions with bounded domain and smooth boundary. To ensure the existence of weak positive solutions first will ensure a positive solution to a classic problem that is the problem eigenvalue p-Laplacian value, and the "linear"problem with zero condition p-Laplacian Dirichelt. Having done this we use the existence of these solutions to ensure that the problem in question admits a weak positive solution via the method of sub-super-solutions.

Sejam M uma variedade de Hadamard com curvatura seccional KM ≤ −k2 < 0 e ∂ M sua fronteira assintótica. Dizemos que M satisfaz a condição de convexidade estrita se, dados x ∈ ∂∞M e W ⊂ ∂∞M aberto relativo contendo x, existe um aberto Ω ⊂ M de classe C2 tais que x ∈ Int (∂ Ω) ⊂ W e M \ Ω ´e convexo. Provamos que a condição de convexidade estrita implica que M éregular no infinito com relação ao operador Q[u] := div a(|∇u|) \ |∇u| ∇u definido no espa¸co de Sobolev W 1,p(M ), onde a ∈ C1([0, +∞)) satisfaz a(0) = 0, at(s) > 0 para todo s > 0, a(s) ≤ C (sp−1 + 1), ∀s ≥ 0, onde C > 0 é uma constante, e a(s) ≥ sq para algum q > 0 e para s ≈ 0 e supomos que é possível resolver problemas de Dirichlet em bolas (compactas) de M com dados contínuos no bordo. Segue disto que sob a condição de convexidade estrita, os problemas de Dirichlet para equação de hipersuperfície mínima e para o p-laplaciano, p > 1, são solúveis para qualquer dado contínuo prescrito no bordo assintótico. Também provamos que se M é rotacionalmente simétrica ou se inf BR+1 KM ≥ −e 2kR /R2+2 , R ≥ R∗, para certos R∗ e E > 0, então M satisfaz a condição de convexidade estrita. / Let M be Hadamard manifold with sectional curvature KM ≤ −k2, k > 0 and ∂∞M its asymptotic boundary. We say that M satisfies the strict convexity condition if, given x ∈ ∂∞M and a relatively open subset W ⊂ 2 ∂∞M containing x, there exists a C open subset Ω ⊂ M such that x ∈ Int (∂∞Ω) ⊂ W and M \ Ω is convex. We prove that the strict convexity condition implies that M is regular at infinity relative to the operator Q [u] := div a(|∇u|) \ |∇u| ∇u , defined on the Sobolev space W 1,p(M ), where a ∈ C 1 ([0, ∞)) satisfies a(0) = 0, at(s) > 0 for all s > 0, a(s) ≤ C (s p−1 + 1), ∀s ≥ 0, where C > 0 is a constant, and a(s) ≥ sq , for some q > 0 and for s ≈ 0 and we suppose that it is possible to solve Dirichlet problems on (compact) balls of M with continuous boundary data. It follows that under the strict convexity condition, the Dirichlet problems for the minimal hypersurface and the p-Laplacian, p > 1, equations are solvable for any prescribed continuous asymptotic boundary data. We also prove that if M is rotationally symmetric or if inf BR+1 KM ≥ −e2kR/R2+2 , R ≥ R∗, for some R∗ and E > 0, then M satisfies the SC condition.

In this work, we study existence and non existence for the Dirichlet problem for the minimal graph equation in non convex domains of the plane. We search for conditions on the boundary data which be the less restricted possible for the solubility of the Dirichlet problem. / Neste trabalho estudamos existência e não existência do problema de Dirichlet para a equação dos gráficos mínimos em domínios não convexos do plano. Procuramos por condições sobre o dado no bordo que sejam as menos restritivas possíveis para que o problema de Dirichlet em questão tenha solução

Sejam M uma variedade de Hadamard com curvatura seccional KM ≤ −k2 < 0 e ∂ M sua fronteira assintótica. Dizemos que M satisfaz a condição de convexidade estrita se, dados x ∈ ∂∞M e W ⊂ ∂∞M aberto relativo contendo x, existe um aberto Ω ⊂ M de classe C2 tais que x ∈ Int (∂ Ω) ⊂ W e M \ Ω ´e convexo. Provamos que a condição de convexidade estrita implica que M éregular no infinito com relação ao operador Q[u] := div a(|∇u|) \ |∇u| ∇u definido no espa¸co de Sobolev W 1,p(M ), onde a ∈ C1([0, +∞)) satisfaz a(0) = 0, at(s) > 0 para todo s > 0, a(s) ≤ C (sp−1 + 1), ∀s ≥ 0, onde C > 0 é uma constante, e a(s) ≥ sq para algum q > 0 e para s ≈ 0 e supomos que é possível resolver problemas de Dirichlet em bolas (compactas) de M com dados contínuos no bordo. Segue disto que sob a condição de convexidade estrita, os problemas de Dirichlet para equação de hipersuperfície mínima e para o p-laplaciano, p > 1, são solúveis para qualquer dado contínuo prescrito no bordo assintótico. Também provamos que se M é rotacionalmente simétrica ou se inf BR+1 KM ≥ −e 2kR /R2+2 , R ≥ R∗, para certos R∗ e E > 0, então M satisfaz a condição de convexidade estrita. / Let M be Hadamard manifold with sectional curvature KM ≤ −k2, k > 0 and ∂∞M its asymptotic boundary. We say that M satisfies the strict convexity condition if, given x ∈ ∂∞M and a relatively open subset W ⊂ 2 ∂∞M containing x, there exists a C open subset Ω ⊂ M such that x ∈ Int (∂∞Ω) ⊂ W and M \ Ω is convex. We prove that the strict convexity condition implies that M is regular at infinity relative to the operator Q [u] := div a(|∇u|) \ |∇u| ∇u , defined on the Sobolev space W 1,p(M ), where a ∈ C 1 ([0, ∞)) satisfies a(0) = 0, at(s) > 0 for all s > 0, a(s) ≤ C (s p−1 + 1), ∀s ≥ 0, where C > 0 is a constant, and a(s) ≥ sq , for some q > 0 and for s ≈ 0 and we suppose that it is possible to solve Dirichlet problems on (compact) balls of M with continuous boundary data. It follows that under the strict convexity condition, the Dirichlet problems for the minimal hypersurface and the p-Laplacian, p > 1, equations are solvable for any prescribed continuous asymptotic boundary data. We also prove that if M is rotationally symmetric or if inf BR+1 KM ≥ −e2kR/R2+2 , R ≥ R∗, for some R∗ and E > 0, then M satisfies the SC condition.