Mathematical Biology Tumor Growth
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Abstract
Abstract
We consider Lipschitz continuous viscosity solutions to evolutive Hamilton-Jacobi equations. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the gradient of the solution. This result is based on a Liouville-type result of classification of global solutions on the half space. Under zero Dirichlet boundary condition, we show that the solution only depends on the normal variable. As a consequence, we show that the existence of a pointwise tangential gradient implies existence of a pointwise normal gradient. For the Liouville-type result, and when the Hamiltonian is not convex, we give a counter-example with a solution which is not one-dimensional. We give two applications. On the one hand, for the classical stationary Dirichlet problem on a bounded domain, we show the existence of a closed subset of the boundary of the domain, where Taylor expansion of the solution is uniform. On the other hand, for Hamilton-Jacobi equations on a network, we show that the space derivative of the solution has a trace at each node, which satisfies a natural germ condition.
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@article{Monneau2026Strictly,
title = {Strictly convex Hamilton-Jacobi equations: strong trace of the gradient},
author = {Régis Monneau},
journal = {ESAIM Control Optimisation and Calculus of Variations},
year = {2026},
doi = {10.1051/cocv/2026057},
url = {https://doi.org/10.1051/cocv/2026057}
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