Two-Phase Anisotropic Free Boundary Problems and Applications to the Bellman Equation in 2D

Caffarelli, Luis A.; Silva, Daniela De; Savin, Ovidiu

doi:10.1007/s00205-017-1198-9

Cited by 12 publications

(11 citation statements)

References 12 publications

(6 reference statements)

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“…It turns out that in dimension n D 2 the results can be improved significantly. This was shown by the authors in collaboration with Caffarelli in [4] where the Lipschitz continuity and the classification of global "purely two-phase" solutions were obtained for linear equations with measurable coefficients under very general free boundary conditions u C D G.u ; ; x/.…”

Section: Introductionmentioning

confidence: 87%

“…The ACF monotonicity formula has been extensively used in various other contexts; however, it is specific to the Laplace operator (see also [5,13]). In [4,8] we investigated the questions of Lipschitz regularity of solutions for two-phase, free boundary problems governed by fully nonlinear operators F .D 2 u/ and with a general isotropic free boundary condition u C D G.u /:…”

Section: Introductionmentioning

confidence: 99%

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Global Solutions to Nonlinear Two‐Phase Free Boundary Problems

Silva

Savin

2019

Comm Pure Appl Math

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Global Solutions to Nonlinear Two‐Phase Free Boundary Problems

Silva

Savin

2019

Comm Pure Appl Math

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“…As mentioned briefly in the beginning, one of the main reasons is that the solutions are not Lipschitz regular in general, since there are homogeneous solutions that are only H€ older continuous, for every dimension larger than or equal to 3. Only recently, Caffarelli, De Silva and Savin [10] proved that the Lipschitz regularity is true in space dimension 2, without the assumption that a þ and a À differ by a scalar multiple. Another reason is that the monotonicity formulae, such as the Alt-Caffarelli-Friedman functional, or balanced-energy functionals of Weiss' type, tend to fail as well, making it difficult to analyse the local properties of the free boundary via global solutions.…”

Section: Introductionmentioning

confidence: 99%

Nodal sets for broken quasilinear partial differential equations with Dini coefficients

Kim

2021

Communications in Partial Differential Equations

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This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation,where a þ and a À are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that a þ and a À are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below H€ older regime, and as a result, we establish a structure theorem for singular sets. We also estimate the Hausdorff measure of nodal sets, provided that the vanishing order of given solution is bounded throughout its nodal set, via an approach that extends the classical argument to certain solutions with discontinuous gradient. Besides, we also prove Lipschitz regularity of solutions and continuous differentiability of their nodal set around regular points.

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“…As mentioned briefly in the beginning, one of the main reasons is that the solutions are not Lipschitz regular in general, since there are homogeneous solutions that are only Hölder continuous, for every dimension larger than or equal to 3. Only recently, Caffarelli, De Silva and Savin [CDS18] proved that the Lipschitz regularity is true in space dimension 2, without such an additional condition. Another reason is that the monotonicity formulae, such as the Alt-Caffarelli-Friedman functional, or balanced-energy functionals of Weiss' type, tend to fail as well, making it difficult to analyse the local properties of the free boundary via global solutions.…”

Section: Introductionmentioning

confidence: 99%

Nodal Sets for Broken Quasilinear Partial Differential Equations with Dini Coefficients

Kim

2020

Preprint

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This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation,where a+ and a− are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that a+ and a− are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below Hölder regime, and as a result, we establish a structure theorem for singular sets. We also estimate the Hausdorff measure of nodal sets, provided that the vanishing order of given solution is bounded throughout its nodal set, via an approach that extends the classical argument to certain solutions with discontinuous gradient. Besides, we also prove Lipschitz regularity of solutions and continuous differentiability of their nodal set around regular points.

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Two-Phase Anisotropic Free Boundary Problems and Applications to the Bellman Equation in 2D

Abstract: Abstract. We prove Lipschitz continuity of solutions to a class of rather general two-phase anisotropic free boundary problems in 2D and we classify global solutions. As a consequence, we obtain C 2,1 regularity of solutions to the Bellman equation in 2D.

Cited by 12 publications

References 12 publications

Global Solutions to Nonlinear Two‐Phase Free Boundary Problems

Global Solutions to Nonlinear Two‐Phase Free Boundary Problems

Nodal sets for broken quasilinear partial differential equations with Dini coefficients

Nodal Sets for Broken Quasilinear Partial Differential Equations with Dini Coefficients

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