Related papers: H\"older regularity for weak solutions of H\"orman…

We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to…

In this paper, we study the H\"older-type interpolation inequality and observability inequality from measurable sets in time for parabolic equations either with L^p unbounded potentials or with electric potentials. The parabolic equations…

This paper is concerned with a class of quasilinear elliptic equations involving some potentials related to the Caffarelli-Korn-Nirenberg inequality. We prove the local boundedness and H\"older continuity of weak solutions by using the…

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…

In 1957, De Giorgi [3] proved the H\"{o}lder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack…

This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport…

We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on…

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$ driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further…

We establish sharp local $C^{1,\alpha}$-regularity for weak solutions to degenerate elliptic equations of $p$-Laplacian type with data in Morrey spaces. The proof relies on the Fefferman-Phong inequality and standard tools from regularity…

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying…

We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect…

In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are H\"{o}lder…

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -\Delta_{p}u -\Delta_{q}u = f(x) u^{-\delta},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*}…

We examine the validity of the Poincar\'e inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

We prove global H\"older regularity for the solutions to the time-harmonic anisotropic Maxwell's equations, under the assumptions of H\"older continuous coefficients. The regularity hypotheses on the coefficients are minimal. The same…

It is shown that the data to solution map for the hyperelastic rod equation is H\"older continuous from bounded sets of Sobolev spaces with exponent $s > 3/2$ measured in a weaker Sobolev norm with index $r < s$ in both the periodic and…

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the H\"older regularity and the Schauder estimates to a class of degenerate parabolic equations of non-divergence form in all dimensions \begin{equation}…

We prove Poincar\'e and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincar\'e inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by…

This article studies the partial H\"older continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t…