
29 Kragujevac J. Math. 31 (2008) 29{42. SOLUTION OF THE DIRICHLET PROBLEM WITH Lp BOUNDARY CONDITION Dagmar Medkov¶a Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo n¶am.13, 121 35 Praha 2, Czech Republic, (e-mail: [email protected]) (Received May 26, 2006) Abstract. The solution of the Dirichlet problem for the Laplace equation is looked for in the form of the sum of a single layer and a double layer potentials with the same density f. The original problem is reduced to the solving of the integral equation with an unknown density f. The solution f of this integral equation is given by the Neumann series. 1. INTRODUCTION This paper is devoted to the Dirichlet problem for the Laplace equation on a Lipschitz domain G ½ Rm with a boundary condition g 2 Lp(@G), where m > 2 and 2 · p < 1. This problem has been studied for years. B. J. E. Dahlberg proved in 1979 that there is a Perron-Wiener-Brelot solution u of this problem, the nontangential maximal function of u is in Lp(@G) and g(x) is the nontangential limit of u for almost all x 2 @G (see [2]). Such solutions have been studied by integral equations method. It was shown for G bounded with connected boundary that for 30 each g 2 Lp(@G) there is f 2 Lp(@G) such that the double layer potential Df with density f is a solution of the Dirichlet problem with the boundary condition g (see [11, 6]). This result does not hold for a general G. If G is unbounded or if the boundary of G is not connected then for each p 2 h2; 1) there is g 2 Lp(@G) such that the solution of the Dirichlet problem with the boundary condition g has not a form of a double layer potential with a density from Lp(@G). We look for a solution in another form. Denote by Sf the single layer potential with density f. We have proved that for every g 2 Lp(@G) there is f 2 Lp(@G) such that Df +Sf is a solution of the Dirichlet problem with the boundary condition g. We look for a solution of the Dirichlet problem in the form Df + Sf. The original problem is reduced to the solving of the integral equation T f = g (see x4). If we look for a solution of the Neumann problem with the boundary condition g in the form Sf we get the integral equation ¿f = g. For G bounded and convex and p = 2 Fabes, Sand and Seo (see [4]) proved that X1 f = ¡2 (2¿ + I)jg j=0 is a solution of the problem ¿f = g. If we look for a solution of the Robin problem ¢u = 0 in G, @u=@n + hu = g in the form of a single layer potential Sf we get the integral equation¿f ~ = g. The following result was proved in [9]: Let @G is locally a C1-deformation of a boundary of a convex set (i.e. for each x 2 @G there are a convex domain D(x) in Rm, a neighbourhood U(x) of x, a coordinate system centred m¡1 at x and Lipschitz functions ª1, ª2 de¯ned on fy 2 R ; jyj < rg, r > 0 such that 1 ª1 ¡ ª2 is a function of class C , (ª1 ¡ ª2)(0;:::; 0) = 0, @j(ª1 ¡ ª2)(0;:::; 0) = 0 0 0 m¡1 0 0 for j = 1; : : : ; m ¡ 1 and U(x) \ @G = f[y ; s]; y 2 R ; jy j < r; s = ª1(y )g, 0 0 m¡1 0 0 U(x) \ @D(x) = f[y ; s]; y 2 R ; jy j < r; s = ª2(y )g), 1 < p · 2, ® > ®0 and g 2 Lp(@G). Then X1 f = ®¡1 (I ¡ ®¡1¿~)jg j=0 is a solution of the equation¿f ~ = g. (Here ®0 depends on h.) Using this result we prove that for G with boundary which is locally C1-deformation of a boundary of a 31 convex domain, 2 · p < 1 and g 2 Lp(@G) the solution f of the equation T f = g, corresponding to the Dirichlet problem with the boundary condition g, is given by X1 f = ®¡1 (I ¡ ®¡1T )jg: j=0 Here 1 1 ® > + kSÂ k 1 2 2 @G L (@G) and Â@G is the characteristic function of @G. 2. FORMULATION OF THE PROBLEM Let a domain G ½ Rm, m > 2, have a compact nonempty boundary @G, which is locally a graph of a Lipschitz function, and @G = @(Rm n cl G). Here cl G denotes the closure of G. It means that for each x 2 @G there is a coordinate system centred at x and a Lipschitz function © in Rm¡1 such that ©(0;:::; 0) = 0 and in some neighbourhood of x the set G lies under the graph of © and Rm n cl G lies above the graph of ©. (We do not suppose that @G is connected.) Then the outward unit normal n(x) to G exists at almost any point x of @G. If x 2 @G, ® > 0, denote the non-tangential approach region of opening ® at the point x ¡®(x) = fy 2 G; jx ¡ yj < (1 + ®) dist(y; @G)g; where dist(y; @G) is the distance of y from @G. If u is a function on G we denote on @G the non-tangential maximal function of u N®(u)(x) = supfu(y); y 2 ¡®(x)g: If c = lim u(y) y!x;y2¡®(x) for each ® > ®0, we say that c is the nontangential limit of u at x. Since G is a Lipschitz domain there is ®0 > 0 such that x 2 cl ¡®(x) for each x 2 @G, ® > ®0. 32 If g 2 Lp(@G), 1 < p < 1, we de¯ne Lp-solution of the Dirichlet problem ¢u = 0 in G; (1) u = g on @G (2) as follows: p Find a function u harmonic in G, such that N®(u) 2 L (@G) for each ® > ®0, u has the nontangential limit u(x) for almost all x 2 @G and u(x) = g(x) for almost all x 2 @G. If G is unbounded require moreover that u(x) ! 0 as jxj ! 1. We will suppose to the end of the paragraph that G is bounded. Let f be a function de¯ned on @G. Denote by ©f the set of all hyperharmonic and bounded below functions u on G such that lim inf u(y) ¸ f(x) y!x;y2G for all x 2 @G. Denote by ªf the set of all hypoharmonic and bounded above functions u on G such that lim sup u(y) · f(x) y!x;y2G for all x 2 @G. Put Hf(x) = inffu(x); u 2 ©fg, Hf(x) = supfu(x); u 2 ªfg. Then Hf · Hf (see [1], Theorem 6.2.5). If Hf = Hf we write Hf = Hf. If Hf = Hf then Hf ´ +1 or Hf ´ ¡1 or Hf is a harmonic function in G (see [1], Theorem 6.2.5). A function f is called resolutive if Hf and Hf are equal and ¯nite-valued. If f is resolutive then Hf is called the PWB-solution (Perron-Wiener- Brelot solution) of the Dirichlet problem with the boundary condition f. If x 2 G then there is a unique probabilistic measure ¹x supported on @G such that Z Hf(x) = f d¹x @G for each resolutive function f (see [1], x6.4). The measure ¹x is called the harmonic measure. Let 1 < p < 1. If G has not boundary of class C1 suppose 2 · p < 1. Let g 2 Lp(@G) and u be a harmonic function in G. Then u is a PWB-solution of the 33 Dirichlet problem with the boundary condition g if and only if u is an Lp-solution of the problem (1)-(2) (see Theorem 4.2). p For 1 < p < 1 and 0 < s < 1 the Sobolev space Ls is de¯ned by p ¡s=2 p m Ls = f(I ¡ ¢) g; g 2 L (R )g: De¯ne µ µ ¶ ¶ Z 1 Z 2 dr 1=2 Ssf(x) = jf(x + ry) ¡ f(x)jdy : 0 jyj<1 r1+2s p p m p m Remark that a function f belongs to Ls if and only if f 2 L (R ) and Ssf 2 L (R ) p p (see [5], Theorem 3.4). De¯ne Ls(G) as the space of restrictions of functions in Ls to G. For 0 < s < 1, 1 < p; q < 1 let us introduce Besov spaces ½ Z 1 ·Z ¸q=p ¾ Bp;q ´ f 2 Lp(Rm); jf(x) ¡ f(x + y)jpdx dy < 1 : s jyjm+ps p;q p;q De¯ne Bs (G) as the space of restrictions of functions in Bs to G. Remark 2.1. Let 2 · p < 1, G be bounded, g 2 Lp(@G). If u is an Lp-solution p p;p of the Dirichlet problem (1), (2) then u 2 L1=p(G) \ B1=p(G). p p Proof. According to [5], Theorem 5.15 there is v 2 L1=p(G) which is an L - solution of the problem (1), (2). The uniqueness of an Lp-solution of the Dirichlet p problem (see [6], Corollary 2.1.6 or [5], Theorem 5.3) gives that u = v 2 L1=p(G). p p;p Since u 2 L1=p(G) we get u 2 B1=p(G) by [5], Theorem 4.1 and [5], Theorem 4.2.
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