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This thesis is split into two parts, the first one is concerned with some problems in stochastic homogenization and the second addresses a problem in singular SPDEs. In the part on stochastic homogenization we are interested in developing large-scale regularity theories for random linear elliptic operators by using estimates for the homogenization error to transfer regularity from the homogenized operator to the heterogeneous one at large scales. In the whole-space case this has been done by Gloria, Neukamm, and Otto through means of a homogenization-inspired Campanato iteration. Here we are specifically interested in boundary regularity and as a model setting we consider random linear elliptic operators on the half-space with either homogeneous Dirichlet or Neumann boundary data. In each case we obtain a large-scale regularity theory and the main technical difficulty turns out to be the construction of a sublinear homogenization corrector that is adapted to the boundary data. The case of Dirichlet boundary data is taken from a joint work with Julian Fischer. In an attempt to head towards a percolation setting, we have also included a chapter concerned with the large-scale behaviour of harmonic functions on a domain with random holes assuming that these are 'well-spaced'. In the second part of this thesis we would like to provide a pathwise solution theory for a singular quasilinear parabolic initial value problem with a periodic forcing. The difficulty here is that the roughness of the data limits the regularity the solution such that it is not possible to define the nonlinear terms in the equation. A well-posedness result, therefore, comes with two steps: 1) Giving meaning to the nonlinear terms and 2) Showing that with this meaning the equation has a solution operator with some continuity properties. The solution theory that we develop in this contribution is a perturbative result in the sense that we think of the solution of the initial value problem as a perturbation of the solution of an associated periodic problem, which has already been handled in a work by Otto and Weber. The analysis in this part relies entirely on estimates for the heat semigroup. The results in the second part of this thesis will be in an upcoming joint work with Felix Otto and Jonas Sauer.

In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure.

The first contribution of this thesis is a new regularity theorem for time harmonic Maxwell's equations with less than Lipschitz complex anisotropic coefficients. By using the L^p theory for elliptic equations, it is possible to prove H¹ and Hölder regularity results, provided that the coefficients are W^1,p for some p = 3. This improves previous regularity results, where the assumption W^1,∞ for the coefficients was believed to be optimal. The method can be easily extended to the case of bi-anisotropic materials, for which a separate approach turns out to be unnecessary. The second focus of this work is the boundary control of the Helmholtz and Maxwell equations to enforce local constraints inside the domain. More precisely, we look for suitable boundary conditions such that the corresponding solutions and their derivatives satisfy certain local non-zero constraints. Complex geometric optics solutions can be used to construct such illuminations, but are impractical for several reasons. We propose a constructive approach to this problem based on the use of multiple frequencies. The suitable boundary conditions are explicitly constructed and give the desired constraints, provided that a finite number of frequencies, given a priori, are chosen in a fixed range. This method is based on the holomorphicity of the solutions with respect to the frequency and on the regularity theory for the PDE under consideration. This theory finds applications to several hybrid imaging inverse problems, where the unknown coefficients have to be imaged from internal measurements. In order to perform the reconstruction, we often need to find suitable boundary conditions such that the corresponding solutions satisfy certain non-zero constraints, depending on the particular problem under consideration. The multiple frequency approach introduced in this thesis represents a valid alternative to the use of complex geometric optics solutions to construct such boundary conditions. Several examples are discussed.

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L_∞(-1; 0; L^{3, β}(B(1) ⋂ ℝ³ ₊)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ³ ₊×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T||v(·, t)||_{L^3,β(ℝ³ ₊)} = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and Šverák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ³, with solenoidal initial data in the critical Besov space ?^-1/4_4,∞(ℝ³), which has certain continuity properties with respect to weak* convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in ℝ³×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T ||v(·, t)||_{L₃(ℝ³)} = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition. / This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition.

[pt] Nesta tese estudamos a extensão da desigualdade fraca de Harnack até o bordo para uma equação de segunda ordem elíptica geral na forma divergência, assumindo pouca regularidade sobre o operador diferencial. Assim, generalizamos e unificamos todos os resultados precedentes deste tipo. Como aplicação, mostramos estimativas a priori para uma classe de problemas elípticos quasilineares com crescimento quadratico no gradiente e investigamos, sob várias hipóteses, a multiplicidade das soluções obtidas para este problema. / [en] This thesis focuses on global extension of the interior weak Harnack inequality for a general class of divergence-type elliptic equations, under very weak regularity assumptions on the differential operator. In this way we generalize and unify all previous results of this type. As an application, we prove a priori estimates for a class of quasilinear elliptic problems with quadratic growth on the gradient and we investigate, under various assumptions, the multiplicity of the solutions obtained for this problem.

[pt] A equação de Isaacs é um exemplo importante de equação elíptica totalmente não-linear, aparecendo em uma grande variedade de disciplinas. Um fato de interesse particular é que tais equações são dirigidas por operadores não convexos. Portanto, são compatíveis com a teoria de EvansKrylov e apresentam delicados desafios quando se trata de sua teoria da regularidade. Descrevemos uma série de resultados recentes sobre a teoria da regularidade da Equação de Isaacs. Estas cobrem estimativas nos espaços Hölder e Sobolev. Argumentamos através de um método genuinamente geométrico, importando informações de uma equação de Bellman relacionada. / [en] Isaacs equation is an important example of fully nonlinear elliptic equation, appearing in a wide of disciplines. Of particular interest is the fact that such equations are driven by nonconvex operators. Therefore, it falls off the scope of the Evans-Krylov theory and poses additional, delicate, challenges when it comes to its regularity theory. We describe a series of recent results on the regularity theory of the Isaacs equation. These cover estimates in Holder and Sobolev spaces. We argue through a genuinely geometrical method, by importing information from a related Bellman equation.

[pt] Neste trabalho examinamos equações completamente não-lineares em dois contextos distintos. A princípio, estudamos jogos de campo médio completamente não-lineares. Aqui, examinamos ganhos de regularidade para as soluções do problema, existência de soluções, resultados de relaxação e aspectos particulares de um example explícito. A segunda metade da tese dedica-se à regularidade ótima das soluções de um modelo completamente não-linear que degenera-se com respeito ao gradiente das soluções. A pergunta fundamental subjacente a ambos os tópicos diz respeito aos efeitos da elipticidade sobre propriedades intrínsecas das soluções de equações não-lineares. Mais precisamente, no caso dos jogos de campo médio, a elipticidade parece magnificada pelos efeitos do acoplamento, enquanto no caso dos problemas degenerados, esta quantidade colapsa em sub-regiões do domínio, dando origem a delicados fenômenos. Nossa análise inclui um breve contexto da inserção do trabalho. / [en] In this thesis, we examine fully nonlinear problems in two distinct contexts. The first part of our work focuses on fully nonlinear mean-field games. In this context, we examine gains of regularity, the existence of solutions, relaxation results, and particular aspects of a one-dimensional problem. The second half of the thesis concerns a (sharp) regularity theory for fully nonlinear equations degenerating with respect to the gradient of the solutions. The fundamental question underlying both topics regards the effects of ellipticity on the intrinsic properties of solutions to nonlinear equations. To be more precise, in the case of mean-field game systems, ellipticity seems to be magnified through the coupling structure. On the other hand, in the degenerate setting, ellipticity collapses, giving rise to intricate regularity phenomena. Our analysis is preceded by some context on both topics.