tailieunhanh - Explicit estimates on a mixed Neumann-Robin-Cauchy problem

The parabolic equation is of divergence form with discontinuous coefficients. We consider a nonlinear condition on a part of the boundary such that the power laws (and the Robin boundary condition) appear as particular cases. | Turk J Math (2016) 40: 1356 – 1373 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Explicit estimates on a mixed Neumann–Robin–Cauchy problem Luisa CONSIGLIERI∗ R. Tom` as da Fonseca 26, Pateo Central 1600-256, Lisbon, Portugal Received: • Accepted/Published Online: • Final Version: Abstract: We deal with the existence of weak solutions for a mixed Neumann–Robin–Cauchy problem. The existence results are based on global-in-time estimates of approximating solutions, and the passage to the limit exploits compactness techniques. We investigate explicit estimates for solutions of the parabolic equations with nonhomogeneous boundary conditions and distributional right-hand sides. The parabolic equation is of divergence form with discontinuous coefficients. We consider a nonlinear condition on a part of the boundary such that the power laws (and the Robin boundary condition) appear as particular cases. Key words: Mixed Neumann–Robin–Cauchy problem, discontinuous coefficient, bounded solution 1. Introduction The existence of solutions to partial differential equations (PDEs) is not sufficient whenever the main objective is their application to other branches of science. In industrial applications, physical fields (such as temperature or potentials) verify PDEs in divergence form with a nondifferentiable leading coefficient. Thus, they do not correspond to the classical solutions. There is a growing demand for the existence of quantitative estimates with explicit constants due to the application of fixed-point arguments [14, 17, 26]. The knowledge of the values of the involved constants in the estimates is crucial. The study of the Dirichlet–Cauchy problem is vast in the literature; see [2, 6, 15, 19–21, 23, 28] to mention a few. It is known that Dirichlet boundary conditions roughly approximate the reality. As a consequence, the study of the Cauchy problem under .