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A regularity result for quasilinear elliptic systems with an equation of energy-type

We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of physics for internal energy or entropy. Relying on the properties of diffusive measures, we prove a Hölder continuity result for these PDE systems in the case of vectorial p-growth and of low regularity features such as mixed boundary conditions on Lipschitz domains. This result is valid in all space dimensions. In the special case where the principal part is linear in the second order, we also study the higher integrability of the gradients and the uniqueness principle in the small. Finally two illustrations in the context of electrothermal coupling are considered, the stationary thermistor model and the Nernst-Planck equations of electrochemistry with variable temperature.