Guerrero Laos, Marilin Nathalya

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The inhomogeneous diffusion equation of Wentzell type with discontinuous data
(2024-07-10) Guerrero Laos, Marilin Nathalya; Rios Soto, Karen; College of Arts and Sciences - Sciences; Vélez Santiago, Alejandro; Romero Oliveras, Juan; Vásquez Urbano, Pedro; Department of Mathematics; Sierra Gil, Heidy
Let Ω ⊂ R^N (N ≥ 3) be a bounded domain with a Lipschitz continuous boundary Γ. In this work, we address the existence and uniqueness of weak solutions for two non-homogeneous diffusion problems in Ω: an elliptic type problem, −Au = f , and a parabolic type problem, ut − Au = f . Here, A is a second-order differential operator with measurable and bounded principal coefficients, not necessarily symmetric, and with measurable and unbounded lower-order coefficients. For both problems, we consider non-homogeneous Wentzell-type boundary conditions on Γ, given by the equation Nu − Bu = g, where Nu represents the conormal derivative of u and B is a second-order operator with similar characteristics to the operator A. Additionally, under minimal assumptions, we obtain a priori estimates for the weak solutions of both problems. For the elliptic problem, these estimates depend on the norms of the data, while for the parabolic problem, the bounds depend on both the norms of the data and the initial condition. These results are fundamental for understanding the behavior and regularity of the solutions under the specific Wentzell-type boundary conditions.