Related papers: Parabolic Regularity in Geometric Variational Anal…

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the…

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…

Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and…

In this note, we give an introduction to the concept of maximal $L^p$-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the…

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We proof Holder regularity in space and time and for translation invariant equations and under different assumptions on the kernels Holder regularity for…

We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic…

A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth…

The adiabatic connection formalism, usually based on the first-order perturbation theory, has been generalized to an arbitrary order. The generalization stems from the observation that the formalism can be derived from a properly arranged…

The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have…

By introducing the notion of distributive constant for a family of closed subschemes, we establish a general form of the second main theorem for algebraic nondegenerate meromorphic mappings from a generalized $p$-Parabolic manifold into a…

The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It…

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent…

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as…

Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and…

Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions…

We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a…

In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity.…

We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral…

A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…