Oblique Derivative Problems For Second Order Linear Elliptic Equations

Due to the important mathematical theory of value and the significance of a wide range of practical applications, elliptic equations, as a main branch of partial differential equations, have been deeply investigated by many mathematicians. Among these, the Schauder theory was established in the early 20th century.For the linear elliptic equation (Ω(?)Rn is bounded)with Dirichlet boundary condition Schauder estimates hold as follow[19]:Suppose and (1) satisfy uniformly elliptic conditions, i.e.0＜λ≤A exists, such that Let is the solution of Dirichlet problem (1), (2) then where C is positive and depends only on n,α,λ,Λ, diamΩ, aij, bi, the Cα(Ω) norm of c and the C2,αnorm of (?)Ω.Using the Schauder estimates, the existence and uniqueness of the classical solution to the Dirichlet problems can be proved. In many practical applications, instead of giving Dirichlet boundary conditions,oblique derivative boundary conditions are prescribed.The paper will introduce the Schauder Theory of the oblique derivative problems for linear elliptic equations.The oblique derivative problem was first posed by Poincare in 1910([23]) where P is an second order elliptic differential operator on Rn,Ωin Rn is bounded,(?) is the unit vector on Rn.When (?)=(?)(the unit outer normal vector on (?)Ω),(3)be-comes Neumann problem([22]).When (?) is everywhere transversal on (?)Ω,(3)becomes an oblique derivative problem,and the existence,uniqueness,regularity conclusions are the same as Neumann problem,except that the existence asks for extra finitely many compati-bility conditions([24]).When (?) is tangent to (?)Ω,the situation of solutions is far more different from former. In[25],if P=△,Ωis a unit ball in R3,(?)=(?)/((?)x),then any harmonic function u=u(x,y,z) matis independent of x,satisfies(3)when f=0,g=0 can generate a finite dimensional null space.From this instance,(?) point in or out ofΩmeans a lot to the value of u.LetΩbe bounded on Rn.For uniformly positive definite function[aij]n×n,vector b(x), scalar c onΩ,vector fieldβof constant length and the scalarγon (?)Ω,let Then the oblique derivative problem can be formulated as followsIn the C2,αtype domain,the a priori estimates to the oblique derivative problem(4) hold as follow([21]):LetΩin Rn be C2,α,u∈C2,α(Ω)be the solution of oblique derivative problem(4),where the normal vectorβv ofβ=(β1,…,βn)is not zero,and |βv|≥χ>0 holds on (?)Ωwithχbeing a constant. Suppose L satisfies f∈Cα(Ω), g∈C1,α(Ω),aij, bi,c∈Cα(Ω),γ,βi,∈C1,α(Ω), and Then where C=C(n,α,λ,Λ,χ,Ω).Using the Schauder estimates, the following existence and uniqueness conclusion holds([21]):Assume that c≤0 inΩandγ(β·v)> 0 additionally, whereγis the unit outer normal on (?)Ω. Then the oblique derivative problem (4) admits uniquely a solution in C2,α(Ω) for all f∈Cα(Ω) and g∈C1,α((?)Ω).In many practical applications, the domain may not belong to C2,α. Therefore, the oblique derivative problems in the domains not belonging to C2,αare also considered. For Lipschitz type domains, the parallel Schauder estimates and the existence and uniqueness of the classical solutions can be established. For nonsmooth domains, although the problems may be ill-posed, the Holder estimates for the solutions still hold.