The Experts below are selected from a list of 4470 Experts worldwide ranked by ideXlab platform
Neshan Wickramasekera - One of the best experts on this subject based on the ideXlab platform.
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a general regularity theory for stable codimension 1 integral varifolds
Annals of Mathematics, 2014Co-Authors: Neshan WickramasekeraAbstract:We give a necessary and sufficient geometric structural condition, which we call the a -Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The a -Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded C 1,a hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral n -varifold on a smooth (n+1) -dimensional Riemannian manifold satisfies the a -Structural Hypothesis for some a?(0,1/2) , its singular set is empty if n=6 , discrete if n=7 and has Hausdorff dimension =n-7 if n=8 ; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a -Structural Hypothesis for some a?(0,1/2) . The a -Structural Hypothesis on an n -varifold for any a?(0,1/2) is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing (n-1) -dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions. An optimal Strong Maximum Principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems
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a sharp Strong Maximum Principle and a sharp unique continuation theorem for singular minimal hypersurfaces
arXiv: Differential Geometry, 2013Co-Authors: Neshan WickramasekeraAbstract:We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and compactness theory for stable minimal hypersurfaces, and on earlier work of Ilmanen, Simon and Solomon--White.
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a general regularity theory for stable codimension 1 integral varifolds
arXiv: Differential Geometry, 2009Co-Authors: Neshan WickramasekeraAbstract:We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the dimension of the varifold is 6 or smaller, discrete if the dimension is 7 and has Hausdorff codimension at least 7 if the dimension is 8 or larger. No initial smallness assumption on the singular set is necessary for these conclusions. The work in particular settles the long standing question, left open by the Schoen-Simon Regularity Theory, as to which weakest size hypothesis on the singular set guarantees the validity of the above conclusions. An optimal Strong Maximum Principle for stationary codimension 1 integral varifolds follows.
Andrea Malchiodi - One of the best experts on this subject based on the ideXlab platform.
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a Strong Maximum Principle for the paneitz operator and a non local flow for the q curvature
Journal of the European Mathematical Society, 2015Co-Authors: Matthew J Gursky, Andrea MalchiodiAbstract:In this paper we consider Riemannian manifolds (M, g) of dimension n ≥ 5, with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a Strong Maximum Principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive Q-curvature.
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a Strong Maximum Principle for the paneitz operator and a non local flow for the q curvature
arXiv: Differential Geometry, 2014Co-Authors: Matthew J Gursky, Andrea MalchiodiAbstract:In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove $(i)$ the Paneitz operator satisfies a Strong Maximum Principle; $(ii)$ the Paneitz operator is a positive operator; and $(iii)$ its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive $Q$-curvature.
Yikan Liu - One of the best experts on this subject based on the ideXlab platform.
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Strong Maximum Principle for multi term time fractional diffusion equations and its application to an inverse source problem
Computers & Mathematics With Applications, 2017Co-Authors: Yikan LiuAbstract:Abstract In this paper, we establish a Strong Maximum Principle for fractional diffusion equations with multiple Caputo derivatives in time, and investigate a related inverse problem of practical importance. Exploiting the solution properties and the involved multinomial Mittag-Leffler functions, we improve the weak Maximum Principle for the multi-term time-fractional diffusion equation to a Stronger one, which is parallel to that for its single-term counterpart as expected. As a direct application, we prove the uniqueness for determining the temporal component of the source term with the help of the fractional Duhamel’s Principle for the multi-term case.
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Strong Maximum Principle for fractional diffusion equations and an application to an inverse source problem
arXiv: Analysis of PDEs, 2015Co-Authors: Yikan Liu, William Rundell, Masahiro YamamotoAbstract:The Strong Maximum Principle is a remarkable characterization of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak Maximum Principle, in this paper we establish a Strong Maximum Principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.
Sergio Polidoro - One of the best experts on this subject based on the ideXlab platform.
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A survey on the classical theory for Kolmogorov equation
arXiv: Analysis of PDEs, 2019Co-Authors: Francesca Anceschi, Sergio PolidoroAbstract:We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a Strong Maximum Principle.
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a geometric statement of the harnack inequality for a degenerate kolmogorov equation with rough coefficients
arXiv: Analysis of PDEs, 2018Co-Authors: Francesca Anceschi, Michela Eleuteri, Sergio PolidoroAbstract:We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a Strong Maximum Principle.
Matthew J Gursky - One of the best experts on this subject based on the ideXlab platform.
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a Strong Maximum Principle for the paneitz operator and a non local flow for the q curvature
Journal of the European Mathematical Society, 2015Co-Authors: Matthew J Gursky, Andrea MalchiodiAbstract:In this paper we consider Riemannian manifolds (M, g) of dimension n ≥ 5, with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a Strong Maximum Principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive Q-curvature.
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a Strong Maximum Principle for the paneitz operator and a non local flow for the q curvature
arXiv: Differential Geometry, 2014Co-Authors: Matthew J Gursky, Andrea MalchiodiAbstract:In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove $(i)$ the Paneitz operator satisfies a Strong Maximum Principle; $(ii)$ the Paneitz operator is a positive operator; and $(iii)$ its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive $Q$-curvature.
