Related papers: $C^{1, \alpha}$-regularity for functions in soluti…

In this paper we establish optimal $C^{1,\alpha}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here…

We prove an $L^p(I,C^\alpha(\Omega))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the…

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…

In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard…

We prove optimal boundary $C^{1,\alpha}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^{\gamma}F(D^2 u)…

We investigate the $C^{1+\alpha}$-regularity of solutions of parabolic equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity solutions…

In this paper, we prove boundary pointwise $C^{k,\alpha}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in…

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian…

In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy…

In this paper, we are interested in obtaining a unified approach for $C^{1,\alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without…

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach…

We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally $C^{1,\gamma}$-regular.

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where…

We present a simple proof of the $C^1$ regularity of $p$-anisotropic functions in the plane for $2\leq p<\infty$. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the…

In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,\alpha}$ ($k\geq 1$, $0<\alpha<1$) regularity with smallness assumptions on the solution…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…

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…

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…

We present a proof of the local H\"older regularity of the horizontal derivatives of weak solutions to the $p$-Laplace equation in the Heisenberg group $\mathbb{H}^1$ for $p>4 $.

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…