Universal inequalities on domains in Euclidean space and their applications

F. G. Avkhadiev

Ufa mathematical journal, Tome 14 (2022) no. 3, pp. 3-16

Résumé

In domains in Euclidean spaces, for test functions, we construct and prove several new Gagliardo-Nirenberg type inequalities with explicit constants. These inequalities are true in any domain, they are nonlinear, integrand functions involve the powers of the absolute values of the gradient and the Laplacian of a test function $u$, as well as factors of type $f(|u(x)|)$, $f'(|u(x)|)$, where $f$ is a continuously differentiable non-decaying function, $f(0)=0$. As weight functions, the powers of the distance from a point to the boundary of the domain serve as well as the powers of the varying hyperbolic (conformal) radius.As applications of universal inequalities of Gagliardo-Nirenberg type we obtain new integral Rellich type inequalities in planar domains with uniformly perfect boundaries. For these Rellich type $L_p$-inequalities we establish criteria of the positivity of the constants, obtain two-sided estimates for these constants depending on the Euclidean maximal modulus of the domain and on the parameter $p\geq 2$. In the proof we use several scalar characteristics for domains with uniformly perfect boundaries.

Détail

Keywords: Gagliardo-Nirenberg type inequality, distance to the boundary, hyperbolic radius, uniformly perfect set.

Bibliographie

[1] E. Gagliardo, “Ulteriori propriet`a di alcune classi di funzioni in pi`u variabili”, Ricerche Mat., 8 (1959), 24-51 | MR | Zbl

[2] L. Nirenberg, “On elliptic partial differential equations”, Ann. Sc. Norm. Super. Pisa. Sci. Fis. Mat. Ser. 3, 13 (1959), 115–162 | MR | Zbl

[3] O.V. Besov, “Integral estimates for differentiable functions on irregular domains”, Sb. Math., 201:12 (2010), 1777–1790 | DOI | DOI | MR | Zbl

[4] V.G. Maz'ya, Sobolev Spaces, Springer, Berlin, 1985 | MR | Zbl

[5] A. Ancona, “On strong barriers and an inequality of Hardy for domains in $\mathbb{R}^n$”, J. London Math. Soc, 34:2 (2), 274–290 | DOI | MR | Zbl

[6] F.G. Avkhadiev, “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31 | MR | Zbl

[7] A.A. Balinsky, W.D. Evans and R.T. Lewis, The Analysis and Geometry of Hardy's Inequality, Universitext, Springer, Heidelberg-New York-Dordrecht-London, 2015 | DOI | MR

[8] F.G. Avkhadiev, “Integral inequalities in domains of hyperbolic type and their applications”, Sb. Math., 206:2 (2015), 1657–1681 | DOI | DOI | MR | Zbl

[9] F.G. Avkhadiev, “Conformally invariant inequalities”, J. Math. Sci., 241:6 (2019), 672–685 | DOI | MR | Zbl

[10] F.G. Avkhadiev, “Euclidean maximum moduli of plane domains and their applications”, Complex Variables and Elliptic Equations, 64:11 (2019), 1869–1880 | DOI | MR | Zbl

[11] M. Ruzhansky, D. Suragan, Hardy Inequalities on Homogeneous Groups, Progress in Mathematics, 327, Birkhauser, 2019 | DOI | MR | Zbl

[12] D.W. Robinson, “Hardy and Rellich inequalities on the complement of convex sets”, J. Aust. Math. Soc., 108:1 (2020), 98–119 | DOI | MR | Zbl

[13] F. Avkhadiev, “Selected results and open problems on Hardy-Rellich and Poincaré-Friedrichs inequalities”, Anal. Math. Phys., 11 (2021), 134, 1–20 | DOI | MR

[14] F.G. Avkhadiev, “Hardy type inequalities involving gradient of distance function”, Ufa Math. J., 13:3 (2021), 3–16 | DOI | MR | Zbl

[15] L.V. Ahlfors, Conformal invariants, Topics in Geometric Function Theory, McGraw - Hill, New-York, 1973 | MR | Zbl

[16] H. Rademacher, “Über partielle und totale Differenzierbarkeit I”, Math. Ann., 89:4 (1919), 340–359 | DOI

[17] H. Federer, Geometric measure theory, Springer, New York, 1969 | MR | Zbl

[18] D.H. Armitage, U. Kuran, “The convexity and the superharmonicity of the signed distance function”, Proc. Amer. Math. Soc., 93:4 (1985), 598–600 | DOI | MR | Zbl

[19] C. Mantegazza, A.C. Mennucci, “Hamilton-Jacobi equations and distance functions on Riemannian manifolds”, Appl. Math. Optim., 47 (2003), 1–25 | DOI | MR

[20] A.I. Nazarov, S.V. Poborchii, Poincaré inequality and its application, St.-Petersburg Publ., St.-Petersburg, 2012 (in Russian)

[21] F.G. Avkhadiev, K.-J. Wirths, Schwarz-Pick Type Inequalities, Birkhäuser Verlag, Basel-Boston-Berlin, 2009 | MR | Zbl

[22] A.F. Beardon, C. Pommerenke, “The Poincaré metric of plane domains”, J. London Math. Soc. (2), 18 (1978), 475–483 | DOI | MR | Zbl

[23] L. Carleson, T.W. Gamelin, Complex dynamics, Springer, New York, 1993 | MR | Zbl

[24] C. Pommerenke, “Uniformly perfect sets and the Poincaré metric”, Arch. Math., 32 (1979), 192–199 | DOI | MR | Zbl

[25] A. Golberg, T. Sugawa, M. Vuorinen, “Teichmüller's theorem in higher dimensions and its applications”, Comput. Methods Funct. Theory., 20:3-4 (2020), 539–558 | DOI | MR | Zbl

[26] F.G. Avkhadiev, “Hardy-Rellich inequalities in domains of the Euclidean space”, J. Math. Anal. Appl., 442 (2016), 469–484 | DOI | MR | Zbl