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VIDEO LIBRARY |
International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
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Estimations of classes of integrals constructed with the help of the classical warping function R. G. Salakhudinov Kazan (Volga Region) Federal University |
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Abstract: Let \begin{gather*} \Delta u =-2 \quad\text{in }G, \\ \begin{alignedat}{2} u&=0 &\qquad &\text{on } \Gamma_0, \\ u&=c_i &\qquad &\text{on } \Gamma_i,\ i=1,\dots,n, \end{alignedat} \end{gather*} where the constants $$ \oint_{\Gamma_i}\frac{\partial u}{\partial n}\, {\mathrm{d}} s=-2a_i,\qquad i=1,\dots,n, $$ In the next two assertions we give estimates for a class of integrals of the warping function. Let a function $$ F(t):=p\int\limits_0^t s^{p-1}f(s){\mathrm{d}} s, $$ where Theorem 1. Let $$ \int_G F(u(x,G))\,{\mathrm{dA}}\le \int_{R_p}F(u(x,R_p))\,{\mathrm{dA}}. $$ $$ \int_G F(u(x,G))\,{\mathrm{dA}}\ge \int_{R_p} F(u(x,R_p))\,{\mathrm{dA}}. $$ Here Using the functionals Theorem 2. Under the assumptions of Theorem 1 the following estimates hold $$ \int_G F(u(x,G))\,{\mathrm{dA}}\le \frac{\mathbf{T}_p(G)}{\mathbf{u}(G)^p}F({\mathbf{u}(G)})-\frac{2\pi {\mathbf{u}(G)}F({\mathbf{u}(G)})}{p+1}+2\pi\int\limits_{0}^{{\mathbf{u}(G)}}F(t)\,{\mathrm{d}} t, $$ where $$ \int_G F(u(x,G))\,{\mathrm{dA}}\ge \frac{\mathbf{T}_p(G)}{{\mathbf{u}(G)}^p}F({\mathbf{u}(G)})-\frac{2\pi {\mathbf{u}(G)}F({\mathbf{u}(G)})}{p+1}+2\pi\int\limits_{0}^{{\mathbf{u}(G)}}F(t)\,{\mathrm{d}} t, $$ here Equalities hold if and only if Language: English |