Lisa Beck

On the Dirichlet problem for variational integrals in BV

Abstract We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral our results extend classical results from the scalar case N = 1—where the problem coincides with the non-parametric least area problem—to the general vectorial setting N ∈ ℕ.

On the Existence of Integrable Solutions to Nonlinear Elliptic Systems and Variational Problems with Linear Growth

We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic radial structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously to the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.

Elliptic regularity theory : a first course

Preliminaries.- Introduction to the Setting.- The Scalar Case.- Foundations for the Vectorial Case.- Partial Regularity Results for Quasilinear Systems.

Globally Lipschitz minimizers for variational problems with linear growth

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space

Partial Regularity Results for Quasilinear Systems

W^{1,1}

Introduction to the Setting

with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler--Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrins paper [19].

The Scalar Case

We next study more general quasilinear systems in divergence form and deal with partial \(C^{1,\alpha }\)-regularity results for their weak solutions. More precisely, we first give a basic higher integrability statement. Then we employ the method of \(\mathcal{A}\)-harmonic approximation, which was introduced in the previous chapter, in order to prove in the first place the partial C 1-regularity of weak solutions outside of a singular set which is of \(\mathcal{L}^{n}\)-measure zero and in the second place the optimal regularity improvement from C 1 to \(C^{1,\alpha }\) for some α > 0 (determined by the regularity of the governing vector field). These results come along with a characterization of the exceptional set on which singularities of a weak solution may arise. However, it does not directly allow for a non-trivial bound on its Hausdorff dimensions, but this requires further work. In different settings, from simple to quite general ones, we explain (fractional) higher differentiability estimates for the gradient of weak solutions. These provide, in turn, the desired bounds for the Hausdorff dimension of the singular set.

Foundations for the Vectorial Case

This chapter contains a short introduction to the concept of weak solutions for partial differential equations of second order in divergence form. We motivate some elementary assumptions (concerning measurability, growth and ellipticity) and then comment on the connection to the minimization of variational functionals, via the Euler–Lagrange formalism.

Regularity versus singularity for elliptic problems in two dimensions

The aim of this chapter is to discuss some full (that is everywhere) regularity results for scalar-valued weak solutions to second order elliptic equations in divergence forms. Weak solutions only belong to some Sobolev space (consequently, neither are they necessarily continuous nor do derivatives a priori exist in the classical sense), and hence, their regularity needs to be investigated. We here prove local Holder regularity of weak solution and present two different (and classical) strategies of proof dating back to the late 1950s. First, we explain De Giorgi’s level set technique, in a unified approach that applies both to weak solutions of elliptic equations and to minimizers of variational integrals, via the study of Q-minimizers of suitable functionals. We then address, for the specific case of linear elliptic equations, an alternative proof of the everywhere regularity result of weak solutions via Moser’s iteration method.

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

In this chapter we continue to investigate the regularity for weak solutions, but now we address the case of vector-valued solutions where we encounter fundamentally new phenomena when compared to the scalar case. In order to concentrate on the central concepts and ideas, we here restrict ourselves to the model case of quasilinear systems that are linear in the gradient variable. We first give two examples of elliptic systems, which admit a discontinuous or even unbounded weak solution. Then we investigate the optimal regularity of weak solutions in dependency of the “degree” of nonlinearity of the governing vector field. In this regard, we start by discussing the linear theory and establish full regularity estimates. This is quite peculiar and a consequence of the particular structure of the coefficients, since for more general systems, as in the counterexamples, one merely expects partial regularity results, that is, regularity outside of negligible sets. Secondly, we present three different strategies for proving partial \(C^{0,\alpha }\)-regularity results for such systems, where the coefficients may depend also explicitly on the weak solution. More precisely, we explain the main ideas for the blow-up technique, the method of \(\mathcal{A}\)-harmonic approximation, and the indirect approach.