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Register a new userOn the second boundary value problem for Lagrangian mean curvature equation
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Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle-Warrens theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.
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In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Holder continuous second derivatives.
We classify regularity for a class of Lagrangian mean curvature type equations, which includes the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is $C^2$ and convex in the gradient variable. We next show that for merely Holder continuous phases, convex solutions are regular if they are $C^{1,beta}$ for sufficiently large $beta$. Singular solutions are given to show that each condition is optimal and that the Holder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.
In this paper we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified {em monodomain model} describing the electric activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in cite{capvoge} in the case of the linear conductivity equation. Finally, we propose some ideas of the reconstruction procedure that might be used to detect the inhomogeneities.
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