A PRIORI ESTIMATE OF THE SOLUTION OF THE DIRICHLET PROBLEM FOR ONE CLASS OF HIGH ORDER DEGENERATING ELLIPTIC EQUATION CONTAINING VARIOUS WEIGHT FUNCTIONS
Assume [Formula: see text] is a planar domain, and [Formula: see text] is a locally bounded distributional solution to the elliptic equation [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text] are real analytic functions defined on [Formula: see text] and the real line [Formula: see text], respectively. We establish asymptotic expansions of [Formula: see text] to arbitrary orders near [Formula: see text], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.
В работе исследуется обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, с переменным потенциалом. Рассматриваемое уравнение выступает модельным уравнением движения во фрактальной среде. Для исследуемого уравнения доказана априорная оценка решения смешанной двухточечной краевой задачи.
We consider an ordinary differential equation of fractional order with the composition of leftand right-sided fractional derivatives, and with variable potential. The considered equation is a model equation of motion in fractal media. We prove an a priori estimate for solutions of a mixed two-point boundary value problem for the equation under study.
On the Solvability of an Evolution Problem with Weighted Integral Boundary Conditions in Sobolev Function Spaces with a Priori Estimate and Fourier’s Method