Related papers: On the best Hoelder exponent for two dimensional e…

We show that for an $L^2$ drift $b$ in two dimensions, if the Hardy norm of $\text{div }b$ is small, then the weak solutions to $\Delta u+b\cdot\nabla u=0$ have the same optimal H\"older regularity as in the case of divergence-free drift,…

In this note we present some uniqueness and comparison results for a class of problem of the form \begin{equation} \label{EE0} \begin{array}{c} - L u = H(x,u,\nabla u)+ h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega), \end{array}…

We consider weak solutions to $$-\Delta_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we…

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order…

We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions…

We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla…

We prove the existence of globally H\"{o}lder continuous solutions to certain elliptic partial differential equations with lower-order terms. Our result is applicable to coefficients controlled by a negative power of the distance from the…

In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*} in $\mathbb{R}^2$, where $a(x)$ and $f_i(x)$ are piecewise H\"older continuous in a…

We complete the study of the regularity for Trudinger's equation by proving that weak solutions are H\"older continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a…

We estimate the Hoelder exponent $\alpha$ of solutions to the Beltrami equation $\dbar f=\mu\de f$, where the Beltrami coefficient satisfies $\|\mu\|_\infty<1$. Our estimate improves the classical estimate $\alpha\ge\|K_\mu\|^{-1}$, where…

We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t} - {\rm div} \left( m |u|^{m-1} \nabla u \right) = f \in L^{q,r}, \quad m >1 ,$$ are locally H\"older continuous, with exponent $$\gamma =\min \left\{…

We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0,…

We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of…

This paper considers the Dirichlet problem $$ -\mathrm{div}(a\nabla u_a)=f \quad \hbox{on}\,\,\ D, \qquad u_a=0\quad \hbox{on}\,\,\partial D, $$ for a Lipschitz domain $D\subset \mathbb R^d$, where $a$ is a scalar diffusion function. For a…

We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form…

We prove interior H\"older estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^{\kappa}\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in (1,\infty)$ and…

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…

This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_{\Omega}\mathcal{H}(x,|\nabla u|)dx \right) \Delta_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \…

We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable…