Related papers: Regularity for the planar optimal p-compliance pro…

In the present paper, we study sharp C^{1;\alpha} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in…

In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an…

This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an…

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$…

In this paper we produce new, optimal, regularity results for the solutions to $p$-Poisson equations. We argue through a delicate approximation method, under a smallness regime for the exponent $p$, that imports information from a limiting…

We show that any minimizer of the well-known ACF functional (for the $p$-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down…

Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive analytic semigroups on UMD-Banach lattices, namely $\ell_p(\ell_q)$ for $p \neq q \in (1, \infty)$,…

We present a proof of the $C^1$ regularity of $p$-orthotropic functions in the plane for $1<p<2$, based on the monotonicity of the derivatives. Moreover we achieve an explicit logarithmic modulus of continuity.

Let $f:\Omega\to\IR^2$ be a mapping of finite distortion, where $\Omega\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(\Omega)$ for some $p>0.$ We establish optimal regularity and area…

In this paper, we consider a kind of degenerate normalized $p$-Laplacian equation with general variable exponents. We establish local $C^{1,\alpha'}$ regularity of viscosity solutions by making use of the compactness argument, scaling…

The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, \alpha}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<\alpha< 1$. The regularity result is…

We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,\alpha}$ estimates on each side of the smooth obstacle, for some small $\alpha > 0$. Our results extend those of Milakis-Silvestre…

In the elliptic theory for $p$-Laplacian-like problems, the H\"{o}lder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting…

For all $1<p<\infty$ and $N\ge 2$ we prove that there is a constant $\alpha(p,N)>0$ such that the $p$-harmonic measure in $\R^N_+$ of a ball of radius $0 < \delta \leq 1$ in $\R^{N-1}$ is bounded above and below by a constant times $\delta…

In this work we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent $\alpha_F$ such that $u$ is $C^{1,\alpha_F}$ on either side of the…

We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,\alpha}$…

We prove regularity and structure results for $p$-elasticae in $\mathbb{R}^n$, with arbitrary $p\in (1,\infty)$ and $n\geq2$. Planar $p$-elasticae are already classified and known to lose regularity. In this paper, we show that every…

We establish global $C^{1,\beta}$ and $W^{2, p}$ regularity for singular Monge-Amp\`ere equations of the form \[\det D^2 u \sim \text{dist}^{-\alpha}(\cdot,\partial\Omega),\quad \alpha\in (0, 1),\] under suitable conditions on the boundary…

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are…