Application of the Almansi formula for constructing polynomial solutions to the Dirichlet problem for a second-order equation
Let \(u\in C^2(\Omega)\) be a classical solution to the second order differential equation with constant coefficients \[ L(D)u(x)\equiv\sum^n_{i,j=1}a_{ji}\frac{\partial^2u}{\partial x_i\partial x_j}=0,\qquad x\in\Omega\subset\mathbb R^n,\tag{1} \] where \(a_{ji}=\)const. and the matrix \(A=(a_{ji})^n_{i,j=1}\) is non-degenerate and symmetric. Assume also that the domain \(\Omega\) has the star property, namely, \(x\in\Omega\) implies \(tx\in\Omega\) for all \(t\in [0,1]\). Introduce a homogeneous polynomial of the second order \(|x|_A^{2}\equiv (A^{-1}x,x)\). Assume \(|x|_A^{2k}\equiv (|x|^2_A)^k\). Then, the author constructs the following system of functions \[ G_k(x;u)=|x|^{2k}_A\int^1_0\frac{(1-\alpha)^{k-1}\alpha^{\frac n2-1}} {4^k k!(k-1)!}u(\alpha x)d\alpha,\qquad k\in\mathbb N, \] and proves that the system satisfies the following conditions \[ L(D)G_1=u,\qquad L(D)G_k=G_{k-1},\quad k\geq 2. \] In addition to (1), the equation \[ L^{m+1}(D)u(x)=0,\qquad x\in\Omega,\tag{2} \] is also studied. Then, the following Almansi representation is valid. Let a function \(p(x)\) be some solution of (2). Then, there exist solutions \(u_0(x),u_1(x),\dots,u_m(x)\) to equation (1) satisfying the Almansi representation \(p(x)=u_0(x)+u_1(x)|x|^2_A+\cdots+u_m(x)|x|_A^{2m}\). This decomposition is then used for constructing a polynomial solution to the Dirichlet problem \[ \begin{aligned} &L(D)u(x)=P(x),\quad x\in\Omega,\\ &u(x)|_{\partial\Omega}=Q(x)|_{\partial\Omega}, \end{aligned} \] where \(P(x)\) and \(Q(x)\) are some polynomials.
scientific article; zbMATH DE number 6108819