### Gradient potential estimates

Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.

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Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space $\mathcal{F}{L}^{s,r}\left(\phantom{\rule{4pt}{0ex}}T\right)$ with $s\ge \frac{1}{2},2<r<4,(s-1)r<-1$ and scaling like ${H}^{\frac{1}{2}-\u03f5}\left(\mathbb{T}\right)$, for small $\u03f5>0$. We also show the invariance of this measure.

Let $\Omega $ be a bounded simply connected domain in the complex plane, $\u2102$. Let $N$ be a neighborhood of $\partial \Omega $, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega $ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega $ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu}$ on $\u2102$ with support contained in $\partial \Omega $ and such that$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\phantom{\rule{0.166667em}{0ex}}\langle \nabla \widehat{u},\nabla \phi \rangle \phantom{\rule{0.166667em}{0ex}}dA=-\int \phi \phantom{\rule{0.166667em}{0ex}}d\widehat{\mu}\end{array}$$whenever $\phi \in {C}_{0}^{\infty}\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega $ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu}$ coincides with harmonic measure...

We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric...

A one-dimensional version of a gradient system, known as “Kobayashi-Warren-Carter system”, is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a “stability” which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further,...

We consider non-linear elliptic equations having a measure in the right-hand side, of the type $div\phantom{\rule{4pt}{0ex}}a(x,Du)=\mu ,$ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation $$-\Delta u+{\mathrm{e}}^{u}({\mathrm{e}}^{u}-1)=\mu \phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega $$ with the Dirichlet boundary condition. Approximating $\mu $ by a sequence ${\left({\mu}_{n}\right)}_{n\in \mathbb{N}}$ of ${L}^{1}$ functions or finite signed measures such that this equation has a solution ${u}_{n}$ for each $n\in \mathbb{N}$, we are interested in establishing the convergence of the sequence ${\left({u}_{n}\right)}_{n\in \mathbb{N}}$ to a function ${u}^{\#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by ${u}^{\#}$.

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$-Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

On donne une borne supérieur du nombre des valeurs propres négatives de l’opérateur de Schrödinger généralisé, cette borne est donnée en fonction d’un nombre fini de cube dyadiques minimaux.