Uniform Poincaré--Sobolev and relative isoperimetric inequalities for classes of domains
- Thomas, Marita
2010 Mathematics Subject Classification
- 46E35 26D10 52A38
- Poincaré-Sobolev inequality, relative isoperimetric inequality, uniform cone property
The aim of this paper is to prove an isoperimetric inequality relative to a d-dimensional, bounded, convex domain &Omega intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(&Omega) of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d).
- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2741--2761.