Related papers: Regularity for the planar optimal p-compliance pro…
Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is…
In this paper, we establish global $C^{1, \alpha}$ regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions. Our work extends the findings in…
Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain…
We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by…
We study approximability of regular constraint satisfaction problems, i.e., CSPs where each variable in an instance has the same number of occurrences. In particular, we show that for any CSP $\Lambda$, existence of an $\alpha$…
We show maximal $L^p$-regularity for non-autonomous Cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation. In particular, on $L^2$, we obtain for all $p \in (1,2]$…
In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,\alpha}$ near regular…
We consider the isoperimetric problem in $\mathbb R^n$ with density for the planar case $n=2$. We show that, if the density is ${\rm C}^{0,\alpha}$, then the boundary of any isoperimetric is of class ${\rm C}^{1,\frac \alpha{3-2\alpha}}$.…
In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Amp\`ere equation. Notably, we…
For a parameter $\gamma\in(1,2)$, we study the fully nonlinear version of the Alt-Phillips equation, $F(D^2u)=u^{\gamma-1}$, for $u\ge 0.$ We establish the optimal regularity of the solution, as well as the $C^1$ regularity of the regular…
We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of…
We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian $(-\Delta)^s$ (and more general integro-differential operators) in the regime $s>\frac{1}{2}$. We prove that once the free boundary…
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior H\"older regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in…
In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<\sigma<2. And we show…
We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary…
We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,\beta}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $\beta \in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the…
We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \begin{align*} \partial_t \big(|u|^{\alpha -1}u \big) - \sum^N_{i=1} \partial_i \big( |\partial_i u|^{p_i - 2}…
Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally…
In this note we are concerned with interior regularity properties of the $p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-)…
In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising…