Related papers: Lipschitz conditions on bounded harmonic functions…

In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.

We prove H\"older regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For example,…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $\Omega\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth…

New sufficient conditions, concerned with the coefficients of harmonic functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$ normalized by $f(0)=h(0)=h'(0)-1=0$, for $f(z)$ to be harmonic close-to-convex functions are…

We prove in an elementary way that for a Lipschitz domain $D\subset \cn$, all plurisubharmonic functions on $D$ can be regularized near any boundary point.

We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose…

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx…

The existence and nonexistence of $\lambda$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then there is…

We consider a function U satisfying a degenerate elliptic equation on (0,+\infty)\times R^N with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain \Omega\subset R^N of class C^{1;1},…

We consider the functional $\int_\Omega g(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(\Omega)$ of the associated Dirichlet problem. We prove that, under…

Let $U$ be a bounded open subset of the complex plane. Let $0<\alpha<1$ and let $A_{\alpha}(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $\alpha$ on the complex plane, are analytic on $U$ and are such…

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…

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance:…

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen…

A Lipschitz space is defined in the Ornstein-Uhlenbeck setting, by means of a bound for the gradient of the Ornstein-Uhlenbeck Poisson integral. This space is then characterized with a Lipschitz-type continuity condition. These functions…

In this short note we consider several widely used L^2-orthogonal Helmholtz decompositions for bounded domains in R^3. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine…

In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be…

Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second…

In trigonometric series terms all polyharmonic functions inside the unit disk are described. For such functions it is proved the existence of their boundary values on the unit circle in the space of hyperfunctions. The necessary and…