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In this manuscript, we investigate regularity estimates for a class of quasilinear elliptic equations in the non-divergence form that may exhibit degenerate behavior at critical points of their gradient. The prototype equation under…
In this paper, we study some regularity issues concerning the gradient of weak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where $\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main results we…
In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla…
Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in…
Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the…
The Dirichlet problem is considered both for degenerate and singular inhomogeneous quasilinear parabolic equations. We prove the existence of a solution $u$ such that $u_t$ belongs to $L_{\infty}$. The $L_{\infty}$ estimate of $u_t$ is…
We generalize the Beurling--Deny--Ouhabaz criterion for parabolic evolution equations governed by forms to the non-autonomous, non-homogeneous and semilinear case. Let $V, H$ are Hilbert spaces such that $V$ is continuously and densely…
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the…
We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…
We prove that bounded solutions to degenerate parabolic double-phase problem modelled upon \[u_t-\dv(|\na u|^{p-2}\na u+a(x,t)|\na u|^{q-2}\na u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,, \] where a nonnegative weight $a$ is $\alpha$-H\"older…
Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of…
We consider positive solutions of the problem \begin{equation} \left\{\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=0\qquad \mbox{in}\;\;\mathbb{R}_+^n,\\ \frac{\partial u}{\partial \nu^a}=u^{q} \qquad \mbox{on}\;\;\partial…
For the $p$-Laplace Dirichlet problem (where $\varphi (t)=t|t|^{p-2}$, $p>1$) \[ \varphi(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>\gamma>0$, while $\int_u^\gamma…
Consider the equation $$ u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ \rho=const >0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in…
We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g…
In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…
The local well-posedness problem is considered for the Dirac-Klein-Gordon system in two space dimensions for data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , where $\|f\|_{\hat{H}^{s,r}} = \| \langle \xi \rangle^s \hat{f}\|_{L^{r'}}$ and…
This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x)…
In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and…
We prove local boundedness and a Harnack inequality for nonnegative weak solutions of the equation $-\nabla\cdot(\mathbf{a}(x)\nabla u)=0$ under a coarse-grained ellipticity assumption on the symmetric coefficient field $\mathbf{a}$.…