Related papers: Hyperbolic regularization effects for degenerate e…
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results…
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary H\"{o}lder regularity under proper…
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and…
This paper extends the Carleman estimates to high dimensional parabolic equations with highly degenerate symmetric coefficients on a bounded domain of Lipschitz boundary and use these estimates to study the controlla?bility the…
We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…
We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T=…
In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…
The paper continues the analysis started in [Cora-Fioravanti-Vita-25,Fioravanti-24] on the local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold. The model…
We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending…
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…
In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…
We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and…
We establish gradient H\"older continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving…
Hein and Pr\"{u}ss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism)…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary,…
In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla…
In this paper local Lipschitz regularity of weak solutions to certain singular elliptic equations involving one-Laplacian is studied. Equations treated here also contains another well-behaving elliptic operator such as $p$-Laplacian with…
In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying…
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 $C^{1,\alpha}$…