Related papers: A Short Proof of H\"older Continuity for Functions…

We give a proof for the H\"older continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak $(1,p)$-Poincar\'e inequality and to satisfy the annular decay…

By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of H\"older continuity of weak solutions of elliptic equations.

We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H\"older continuous. Some applications are given. As a first application, we give a regularity…

In this paper, we introduce a new class of De Giorgi type functions, denoted by \(\mathcal{B}_{G(x,t)}\), and establish the H\"older continuity of its elements under suitable additional assumptions on the generalized \textnormal{N}-function…

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order $p$. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex…

We study H\"older continuity for solutions of the $p$-Laplace equation. This is established through a method involving an ordinary differential inequality, in contrast to the classical proof of the De Giorgi-Nash-Moser Theorem which uses…

We obtain a generalization of the DeGiorgi Lemma to the infinitely degenerate regime and apply it to obtain continuity of weak solutions to certain infinitely degenerate equations. This reproduces the continuity result obtained in…

We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are H\"older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and…

Local H\"older regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling…

We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and H\"older continuity with an explicit…

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and…

We give an alternative proof for H\"older regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by…

Local oscillation of a function satisfying a H\"older condition is considered and it is proved that its growth is governed by a version of the Law of the Iterated Logarithm.

In this paper we study the everywhere H\"oder continuity of the minima of a class of vectorial integral funcionals

In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Amp\`ere operator. We provide necessary and sufficient conditions for the problem to have H\"older continuous solutions.

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

The aim of this article is to present a concise proof for the interior H\"{o}lder continuity of two-dimensional stationary Q-valued maps. The proof employs a blow-up argument based on the monotonicity formula for the frequency function,…

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a H\" older continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a…

In this paper we study the everywhere H\"older continuity of the minima of a class of vectorial integral functionals