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Isoperimetric characteristics of domains and variational inequalities of mathematical physics

This paper explores isoperimetric characteristics of domains within noncompact Riemannian manifolds, focusing on variational inequalities significant in mathematical physics. It presents fundamental concepts established by V.M. Miklyukov and M.-K. Vuorinen, highlighting the impact of isoperimetric profiles and their constants. The work delves into specific cases, such as Hardy's inequality in planar strips, and compares constants in various scenarios. The results are contextualized within established theorems and inequalities, emphasizing their applications in geometric analysis and mathematical physics.

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Isoperimetric characteristics of domains and variational inequalities of mathematical physics

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  1. Isoperimetric characteristics of domains and variational inequalities of mathematical physics IlnarShafigullin (Kazan State University)

  2. Basic concepts by V.M.Miklyukov and M.-K.Vuorinen. Letbe an n-dimensional connected noncompact Riemannian С2-manifold without boundary. For where

  3. Isoperimetric profile

  4. The constant which characterizes isoperimetric profile

  5. Theorem (V.M.Miklyukov and M.-K.Vuorinen, 1999) Let . If has an isoperimetric profile with B<∞, then for any function

  6. S.T. Yau result (1975) Let be Riemannian manifold, with Gaussian curvature K and k=const<0.

  7. Case when the domain is the strip on the plane

  8. Case when the domain is the strip on the plane

  9. Case when the domain is the strip on the plane

  10. Hardy inequality for the strip

  11. Comparison of constants Bin some special cases when p=q

  12. References • Ф.Г. Авхадиев, Неравенства для интегральных характеристик областей, Учебное пособие, Казань, КГУ, 2006 – 142 с. • V.M.Miklyukov, M.K.Vuorinen, Hardy’s inequality for -functions on riemanian manifolds // Proc. Amer. Math. Soc. 127, №9 (1999) 2145-2154. • F.G. Avkhadiev, K.J. Wirths, Unified Poincare and Hardy inequality sharp constants for convex domains, Z.Angev.Math.Meth. 87, No 8-9, 632-642 (2007) • S.T.Yau, Isoperimetric constants and the first eigenvalue of a compact Riemanian manifold, Ann.Sci. Ecole Norm. Sup. (4) 8 (1975), 487-507. • В.М.Миклюков, Неравенство Харди для функций с обобщенными производными на римановых многообразиях. Trudy IPMM NAN Ukrainy. v. 3. 1998. c. 158-172.

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