Multivalued nonmonotone dynamic boundary condition

Boundary value problems

Sultana Ben Aadi¹  Khadija Aayadi¹  Khalid Akhlil¹  Mourad El Ouali¹ 

[1] Polydisciplinary Faculty of Ouarzazate, Ibn Zohr University, Agadir, Morocco;

关键词: Dynamic boundary hemivariational inequality;    Wentzell boundary condition;    Clarke subdifferential;    Nonconvex optimization;    47H04;    47H14;    47H30;    47J20;    47J22;    47J35;   

DOI  :  10.1186/s13661-021-01517-6

来源: Springer

【 摘 要 】

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

【 授权许可】

CC BY