Diffuse measures and nonlinear parabolic equations

By Francesco Petitta, Augusto C. Ponce, Alessio Porretta. 2011

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AbstractGiven a parabolic cylinder Q = (0, T) × Ω, where $${\Omega\subset \mathbb {R}^N}$$ is a bounded domain, we prove new properties of solutions of $$u_t-\Delta_p u = \mu \quad \text{in }Q$$with Dirichlet boundary conditions, where μ is a finite Radon measure in Q. We first prove a priori estimates on the p-parabolic capacity of level sets of u. We then show that diffuse measures (i.e., measures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures μk = (Tk(u))t−Δp(Tk(u)), and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of $$u_t-\Delta_{p} u + h(u)=\mu \quad \text{in }Q,$$for any function h such that h(s)s ≥ 0 and for any diffuse measure μ; when h is nondecreasing, we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/303d08f3eaef418c63e7b8d630fd7265d0eb93d6

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Local and global properties of solutions of quasilinear Hamilton-Jacobi equations (2014, background): “By a variant of the results in [6],[7] due to [33] it follows that the regularity statements (3.”