A gradient estimate for nonlocal minimal graphs
| Autor: | Xavier Cabré, Matteo Cozzi |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. EDP - Equacions en Derivades Parcials i Aplicacions |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Measurable function General Mathematics gradient estimates weak Harnack inequalities regularity results 01 natural sciences Sobolev inequality 58J05 Mathematics - Analysis of PDEs 0103 physical sciences FOS: Mathematics Ball (mathematics) 0101 mathematics 47G20 fractional Sobolev inequalities Harnack's inequality Mathematics Minimal surface Mean curvature Jacobi operator 010102 general mathematics 53A10 nonlocal minimal surfaces Matemàtiques i estadística [Àrees temàtiques de la UPC] Differential equations Partial Equacions diferencials parcials 49Q05 35J60 nonlocal minimal graphs Differential Geometry (math.DG) Bounded function rigidity theorems 28A75 010307 mathematical physics Analysis of PDEs (math.AP) |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) Recercat. Dipósit de la Recerca de Catalunya instname Duke Math. J. 168, no. 5 (2019), 775-848 |
| Popis: | We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^\infty$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n = 1, 2$ (but without a quantitative bound), in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi & Miranda for solutions of the classical mean curvature equation. To appear in Duke Math. J |
| Databáze: | OpenAIRE |
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