To the Weak Solvability of Dirichlet Problem for a Fractional Order Degenerate Elliptic Equation
F.I. Mamedov, N.M. Mammadzada, S.M. Mammadli
Abstract. In this paper, we study the weak solvability of the
nonhomogeneous Dirichlet problem for a degenerate fractional order
elliptic equation:
\({( - \mathrm{\Delta})}^{\frac{\alpha}{2}}\ \left( \omega(x){( - \mathrm{\Delta})}^{\frac{\alpha}{2}}u \right) = f(x),\ \ \ x \in \Omega \subset R^{n},\ \ \ \alpha \in (0;1)\)
\textbar{} \(u|_{R^{n}\backslash\Omega} = \varphi(x)\). For that a
suficient condition is found on the datas of problem
\(\Omega,\ \alpha,\ n,\ \) the weight function
\(\omega:\ \mathbb{R}^{n} \rightarrow \lbrack 0,\ \infty)\) and the
functions \(f,\ \varphi.\) To the proof, a weighted fractional order
Sobolev-Poincare type inequality, and the Lax-Milgram principle is
applied. References. [1] M. Azouzi, L.Guedda Existence result for nonlocal boundary value problem of fractional order at resonance with p-Laplacian operator, Azerb. J. of Math, 13(1), 2023, 14–33.[2] A.M. Caetano, Approximation by functions of compact support in BesovTriebel-Lizorkin spaces on irregular domains, Studia Math, 142(1), 2000, 47–63. [3] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Dif. Equ, 32, 2007,1245–1260. [4] B. Dyda, M. Kijaczko On density of compactly supported smooth functions in fractional Sobolev spaces, Annali di Matem. Pura ed Appl., 201, 2022, 1855– 1867. [5] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [6] K.H. Karlsen, F. Petitta, S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publ. Mat. , 55, 2011, 151–161. [7] N.S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), Switzerland, Springer,180, 1973. [8] G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math., Amer. Math. Soc., Providence, RI, 105,2009. [9] F.I. Mamedov, N.M. Mammadzada, L.E. Persson, A new fractional order Poincare’s inequality with weights, Math. Ineq. Appl., 23(2), 611–624, 2020. [10] F.I. Mamedov, On the multidimensional weighted Hardy inequalities of fractional order, Proc. Inst. Math. Mech. Acad. Sci. Azerb, 10(XVII), 102–114, 1999. [11] F.I. Mamedov, V.A. Mamedova, On Sobolev-Poincare-Friedrichs type weight inequalities, Azerb.J.of Math. 12(2), 92–108, 2022 [12] F.I. Mamedov, S.M. Mammadli, On the fractional order weighted Hardy inequality for monotone functions, Proc. Inst. Math. Mech. Azerb, 42(2), 257– 264, 2016. [13] V. Maz’ya, T. Shaposhnikova, Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), 337, Springer-Verlag, Berlin, 2009. [14] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Des Sci. Math., 136(5), 521–573, 2012. [15] A. Niang, Fractional Elliptic Equations, African Institute for Mathematical Sciences (AIMS), Senegal, Master Thesis, 2014. [16] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl., Walter de Gruyter Co., Berlin, 3, 1996. [17] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., Princeton University Press, Princeton, NJ, 30, 1970. [18] H. Triebel, Interpolation Theory, Fourier Analysis and Function Spaces, Teubner-Texte Math., Teubner, Leipzig, 1977. [19] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital., Springer-Verlag, Berlin, Heidelberg, 3, 2007. |
|