Year: 2021
Type: article
Abstract: We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region $$\varOmega \subseteq {\mathbb {R}}^n$$ Ω⊆Rn . We will deal with the energy functional $${\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )$$ F(v,E):=∫Ω[F(∇v)+1EG(∇v)+fE(x,v)]dx+P(E,Ω) . The bulk energy depends on a functionvand its gradient $$\nabla v$$ ∇v . It consists in two strongly quasi-convex functionsFandG, which have polinomialp-growth and are linked with theirp-recession functions by a proximity condition, and a function $$f_E$$ fE , whose absolute valuesatisfies aq-growth condition from above. The surface penalization term is proportional to the perimeter of a subsetEin $$\varOmega $$ Ω . The term $$f_E$$ fE is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with $${\mathscr {F}}$$ F . The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove thatuis locally Hölder continuous andAis equivalent to an open set $${\tilde{A}}$$ A~ . We finally get $$P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega $$ P(A,Ω)=Hn-1(∂A~∩Ω ).
Author: Lorenzo Lamberti
Institution: University of Salerno
Language: Latvian
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,780
OpenAlex ID: W3153896420