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Gradient potential estimates

Giuseppe Mingione (2011)

Journal of the European Mathematical Society

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

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani (2012)

Journal of the European Mathematical Society

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 L s , r ( T ) with s 1 2 , 2 < r < 4 , ( s - 1 ) r < - 1 and scaling like H 1 2 - ϵ ( 𝕋 ) , for small ϵ > 0 . We also show the invariance of this measure.

p Harmonic Measure in Simply Connected Domains

John L. Lewis, Kaj Nyström, Pietro Poggi-Corradini (2011)

Annales de l’institut Fourier

Let Ω be a bounded simply connected domain in the complex plane, . Let N be a neighborhood of Ω , let p be fixed, 1 &lt; p &lt; , and let u ^ be a positive weak solution to the p Laplace equation in Ω N . Assume that u ^ has zero boundary values on Ω in the Sobolev sense and extend u ^ to N Ω by putting u ^ 0 on N Ω . Then there exists a positive finite Borel measure μ ^ on with support contained in Ω and such that | u ^ | p - 2 u ^ , φ d A = - φ d μ ^ whenever φ C 0 ( N ) . If p = 2 and if u ^ is the Green function for Ω with pole at x Ω N ¯ then the measure μ ^ coincides with harmonic measure...

Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

Tomasz Klimsiak, Andrzej Rozkosz (2016)

Colloquium Mathematicae

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...

Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system

Hiroshi Watanabe, Ken Shirakawa (2014)

Mathematica Bohemica

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,...

The Calderón-Zygmund theory for elliptic problems with measure data

Giuseppe Mingione (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider non-linear elliptic equations having a measure in the right-hand side, of the type div a ( x , D u ) = μ , 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.

The mean curvature measure

Quiyi Dai, Neil S. Trudinger, Xu-Jia Wang (2012)

Journal of the European Mathematical Society

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...

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

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.

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