This paper is in relation with a Note of Comptes Rendus de lAcademie des Sciences 2005. We have an idea about a lower bounds of sup+inf (2 dimensions) and sup*inf (dimensions >2).
We give a derivation for the value of inf-sup constant for the bilinear form (p, div u). We prove that the value of inf-sup constant is equal to 1.0 in all cases and is independent of the size and shape of the domain. Numerical tests for validation o
f inf-sup constants is performed using finite dimensional spaces defined in cite{2020jain} on two test domains i) a square of size $Omega = [0,1]^2$, ii) a square of size $Omega = [0,2]^2$, for varying mesh sizes and polynomial degrees. The numeric values are in agreement with the theoretical value of inf-sup term.
In this paper, we establish compactness and existence results to a Branson-Paneitz type problem on a bounded domain of R^n with Navier boundary condition.
We consider parabolic systems with nonlinear dynamic boundary conditions, for which we give a rigorous derivation. Then, we give them several physical interpretations which includes an interpretation for the porous-medium equation, and for certain re
action-diffusion systems that occur in mathematical biology and ecology. We devise several strategies which imply (uniform)}$L^{p} and}$L^{infty}$ estimates on the solutions for the initial value problems considered.