Existence and uniqueness of classical solutions for certain degenerated elliptic equations of the second order
Toshio Horiuchi · Kyoto journal of mathematics · 1984
IntroductionLet Q be a bounded domain in R ' w ith sm ooth boundary.The operators d which we shall treat in this memoire are of second order, linear, elliptic in the interior of S2 and degenerated only in normal direction at each point of the boundary.U nder some assumptions on d , the existence and uniqueness of the classical solution u of the equation d u = f will be shown for any given function f with certain Holder con- tinuity up to the boundary.W e impose no boundary condition because we assume the -entrance property -of the boundary with respect to d .There are many authors who have studied various types of degenerated elliptic e q u a tio n s.Baouendi [2] treated the equations degenerated at th e boundary, but for which the boundary is non-characteristic.studied the equations degenerated in all directions a t the b o u n d a ry . Them ain tool in these two works is the elliptic regularization initiated by Oleinik (see Oleinik-Radkevic [7]) and the theory of interpolation in L 2 framework.Recently, Goulaouic-Shimakura [6] studied the same class o f operators as in [3 ] in th e W ilder spaces.A nd G raham [10] studied the Dirichlet problems for Bergman Laplacian also in some W ilder spaces.O ur interest in this memoire is to study the same type of operators as in the Chapter V of Graham 's article.But the Wilder spaces with which we work are not the same because of the difference of the boundary conditions.O ur method is, as in [6] and [10], to m ake use of the elementary solution for the simplest model of our operators.In §1, we consider the model LOE in the half-space, and explain the non-isotropic degeneracy at the b o u n d a ry .In §2, we describe the general setting of our equations in a bounded dom ain, and state the m ain results.In th is w o r k , s o m e a priori inequalities o f Schauder type for solutions are essential.In we reduce these inequalities to the case of the half-space.And the a priori inequalities in the halfspace are finally established in § 5 .The §4 is devoted to introduce the elementary solutions of L a n d 1,,+,1.T h e results on the existence and uniqueness of the