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Wenhui Shi  One of the best experts on this subject based on the ideXlab platform.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
Annales De L Institut Henri Poincareanalyse Non Lineaire, 2017CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:Abstract This article deals with the variable coefficient thin obstacle problem in n + 1 dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and W 1 , p metrics with p ∈ ( n + 1 , ∞ ] . In this framework, we prove the C 1 , α Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal C 1 , min { 1 − n + 1 p , 1 2 } Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in [18] and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for W 1 , p metrics and W 2 , p obstacles with p ∈ ( 2 ( n + 1 ) , ∞ ] .

the variable coefficient thin obstacle problem higher Regularity
arXiv: Analysis of PDEs, 2016CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:In this article we continue our investigation of the thin obstacle problem with variable coefficients which was initiated in \cite{KRS14}, \cite{KRSI}. Using a partial HodographLegendre transform and the implicit function theorem, we prove higher order H\"older Regularity for the regular free boundary, if the associated coefficients are of the corresponding Regularity. For the zero obstacle this yields an improvement of a \emph{full derivative} for the free boundary Regularity compared to the Regularity of the metric. In the presence of nonzero obstacles or inhomogeneities, we gain \emph{three halves of a derivative} for the free boundary Regularity with respect to the Regularity of the inhomogeneity. Further we show analyticity of the regular free boundary for analytic metrics. We also discuss the low Regularity setup of $W^{1,p}$ metrics with $p>n+1$ with and without ($L^p$) inhomogeneities. Key new ingredients in our analysis are the introduction of generalized H\"older spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the BaouendiGrushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) Regularity and the splitting technique from \cite{KRSI}.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
arXiv: Analysis of PDEs, 2015CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,\alpha}$ Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1\frac{n+1}{p}, \frac{1}{2}\}}$ Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.
Herbert Koch  One of the best experts on this subject based on the ideXlab platform.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
Annales De L Institut Henri Poincareanalyse Non Lineaire, 2017CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:Abstract This article deals with the variable coefficient thin obstacle problem in n + 1 dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and W 1 , p metrics with p ∈ ( n + 1 , ∞ ] . In this framework, we prove the C 1 , α Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal C 1 , min { 1 − n + 1 p , 1 2 } Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in [18] and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for W 1 , p metrics and W 2 , p obstacles with p ∈ ( 2 ( n + 1 ) , ∞ ] .

the variable coefficient thin obstacle problem higher Regularity
arXiv: Analysis of PDEs, 2016CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:In this article we continue our investigation of the thin obstacle problem with variable coefficients which was initiated in \cite{KRS14}, \cite{KRSI}. Using a partial HodographLegendre transform and the implicit function theorem, we prove higher order H\"older Regularity for the regular free boundary, if the associated coefficients are of the corresponding Regularity. For the zero obstacle this yields an improvement of a \emph{full derivative} for the free boundary Regularity compared to the Regularity of the metric. In the presence of nonzero obstacles or inhomogeneities, we gain \emph{three halves of a derivative} for the free boundary Regularity with respect to the Regularity of the inhomogeneity. Further we show analyticity of the regular free boundary for analytic metrics. We also discuss the low Regularity setup of $W^{1,p}$ metrics with $p>n+1$ with and without ($L^p$) inhomogeneities. Key new ingredients in our analysis are the introduction of generalized H\"older spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the BaouendiGrushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) Regularity and the splitting technique from \cite{KRSI}.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
arXiv: Analysis of PDEs, 2015CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,\alpha}$ Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1\frac{n+1}{p}, \frac{1}{2}\}}$ Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.
Ruijun Wu  One of the best experts on this subject based on the ideXlab platform.

partial Regularity for a nonlinear sigma model with gravitino in higher dimensions
Calculus of Variations and Partial Differential Equations, 2018CoAuthors: Jurgen Jost, Ruijun WuAbstract:We study the Regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler–Lagrange equations and consider the Regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial Regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension $$<6$$ , partial Regularity holds for stationary solutions with respect to general gravitino fields.

partial Regularity for a nonlinear sigma model with gravitino in higher dimensions
Calculus of Variations and Partial Differential Equations, 2018CoAuthors: Jurgen Jost, Ruijun WuAbstract:We study the Regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler–Lagrange equations and consider the Regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial Regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension $$<6$$ , partial Regularity holds for stationary solutions with respect to general gravitino fields.

partial Regularity for a nonlinear sigma model with gravitino in higher dimensions
arXiv: Differential Geometry, 2017CoAuthors: Jurgen Jost, Ruijun WuAbstract:We study the Regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the EulerLagrange equations and consider the Regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial Regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension less than 6, partial Regularity holds for stationary solutions with respect to general gravitino fields.
Soravia Pierpaolo  One of the best experts on this subject based on the ideXlab platform.

Regularity of the minimum time and of viscosity solutions of degenerate eikonal equations via generalized Lie brackets
2020CoAuthors: Bardi Martino, Feleqi Ermal, Soravia PierpaoloAbstract:In this paper we relax the current Regularity theory for the eikonal equation by using the recent theory of { setvalued} iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hoelder Regularity of the minimum time function in optimal control. We then apply such result to prove H\"older continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low Regularity of the coefficients. We also prove that the sufficient conditions for the H\"older Regularity are essentially necessary, at least for smooth vector fields and target

Regularity of the Minimum Time and of Viscosity Solutions of Degenerate Eikonal Equations via Generalized Lie Brackets
'Springer Science and Business Media LLC', 2020CoAuthors: Bardi Martino, Feleqi Ermal, Soravia PierpaoloAbstract:In this paper we relax the current Regularity theory for the eikonal equation by using the recent theory of setvalued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Ho \u308lder Regularity of the minimum time function in optimal control. We then apply such result to prove Ho \u308lder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low Regularity of the coefficients. We also prove that the sufficient conditions for the Ho \u308lder Regularity are essentially necessary, at least for smooth vector fields and target
Angkana Ruland  One of the best experts on this subject based on the ideXlab platform.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
Annales De L Institut Henri Poincareanalyse Non Lineaire, 2017CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:Abstract This article deals with the variable coefficient thin obstacle problem in n + 1 dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and W 1 , p metrics with p ∈ ( n + 1 , ∞ ] . In this framework, we prove the C 1 , α Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal C 1 , min { 1 − n + 1 p , 1 2 } Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in [18] and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for W 1 , p metrics and W 2 , p obstacles with p ∈ ( 2 ( n + 1 ) , ∞ ] .

the variable coefficient thin obstacle problem higher Regularity
arXiv: Analysis of PDEs, 2016CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:In this article we continue our investigation of the thin obstacle problem with variable coefficients which was initiated in \cite{KRS14}, \cite{KRSI}. Using a partial HodographLegendre transform and the implicit function theorem, we prove higher order H\"older Regularity for the regular free boundary, if the associated coefficients are of the corresponding Regularity. For the zero obstacle this yields an improvement of a \emph{full derivative} for the free boundary Regularity compared to the Regularity of the metric. In the presence of nonzero obstacles or inhomogeneities, we gain \emph{three halves of a derivative} for the free boundary Regularity with respect to the Regularity of the inhomogeneity. Further we show analyticity of the regular free boundary for analytic metrics. We also discuss the low Regularity setup of $W^{1,p}$ metrics with $p>n+1$ with and without ($L^p$) inhomogeneities. Key new ingredients in our analysis are the introduction of generalized H\"older spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the BaouendiGrushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) Regularity and the splitting technique from \cite{KRSI}.

the variable coefficient thin obstacle problem optimal Regularity and Regularity of the regular free boundary
arXiv: Analysis of PDEs, 2015CoAuthors: Herbert Koch, Angkana Ruland, Wenhui ShiAbstract:This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary Regularity, the behavior of the solution close to the free boundary and the optimal Regularity of the solution in a low Regularity setup. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,\alpha}$ Regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1\frac{n+1}{p}, \frac{1}{2}\}}$ Regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of nonzero and interior thin obstacles. We obtain the optimal Regularity of the solutions and the Regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.