Giraud conditions

Conditions for the solvability in the classical sense of the fundamental boundary value problems for second-order linear elliptic equations. Suppose that the following elliptic equation is given in a bounded $N$-dimensional domain $D$ ($N\geq2$) with boundary $\Gamma$:

$$Lu\equiv\sum_{i,j=1}^Na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j}+\sum_{i=1}^Nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u=f(x).\tag{*}$$

It is required to find a function $u$ that: 1) lies in the class $C^2(D)\cap C^0(D+\Gamma)$; 2) satisfies condition \ref{*} in $D$; and 3) satisfies the condition $u=\phi$ on $\Gamma$ (the first boundary value problem, or Dirichlet problem), or the condition

$$\frac{\partial u(x)^+}{\partial\nu}+\beta(x)u(x)=\phi(x)$$

(the second boundary value problem, or Neumann problem), or else the condition

$$\frac{\partial u(x)^+}{\partial l}+\beta(x)u(x)=\phi(x)$$

(the third boundary value problem). Here $\nu$ is the direction of the conormal, with direction cosines

$$\cos(\nu,x_i)=\frac1a\sum_{j=1}^Na_{ij}(x)\cos(n,x_j),\quad i=1,\dots,N,$$

$$a=\left[\sum_{i=1}^N\left(\sum_{j=1}^Na_{ij}\cos(n,x_j)\right)^2\right]^{1/2},$$

$n$ is the outward normal to $\Gamma$, $l$ is any direction such that $\cos(l,n)\geq\delta>0$ for all $x\in\Gamma$, and the sign $+$ means that the limit value is taken from within $D$.

The Giraud conditions for the solvability of the boundary value problems indicated consist of the following. If the coefficients $a_{ij}$, $b_i$, $c$ of $L$ are of class $C^{0,\mu}$ in $(D+\Gamma)$, if the right-hand side $f\in C^{0,\mu}(D)\cap C^0(D+\Gamma)$, if the boundary conditions $\phi+C^0(\Gamma)$ (for the second and third boundary value problems), if $\cos(l,x_i)\in C^{0,\mu}(\Gamma)$ (for the third boundary value problem), and if the boundary $\Gamma$ of $D$ is of class $A^{l,\mu}$, then the Fredholm alternative holds for the first, second and third boundary value problems. That is, either the corresponding homogeneous problem has only the trivial solution, and then the inhomogeneous problem has a unique solution for arbitrary $f$ and $\phi$, or else the homogeneous problem has $p$ linearly independent solutions $u_1,\dots,u_p$, $0<p<\infty$, and then the inhomogeneous problem has a solution only if $p$ linear functionals in $f$ and $\phi$ vanish. If this last condition is satisfied, the inhomogeneous problem has infinitely-many solutions, and if $u_0$ is one of them, then the general solution can be written in the form $u_0+\sum_{i=1}^pc_iu_i$, where the $c_i$ are arbitrary constants. In the case where the coefficients of $L$ are smoother ($a_{ij}\in C^{2,\mu}$, $b_i\in C^{l,\mu}$), so that the adjoint operator $L^*$ can be considered, the requirement that the linear functionals in $f$ and $\phi$ vanish reduces to the orthogonality of $f$ and $\phi$ with all $p$ linearly independent solutions of the adjoint homogeneous problem. The conditions were obtained by G. Giraud [1]–[3].

References

[1]	G. Giraud, "Existence de certaines dérivées des functions de Green; consequences pour les problèmes du type de Dirichlet" C.R. Acad. Sci. Paris , 202 (1936) pp. 380–382
[2a]	G. Giraud, "Généralisation des problèmes sur les opérateurs du type elliptique" Bull. Sci. Math. , 56 (1932) pp. 248–272
[2b]	G. Giraud, "Généralisation des problèmes sur les opérateurs du type elliptique" Bull. Sci. Math. , 56 (1932) pp. 281–312
[2c]	G. Giraud, "Généralisation des problèmes sur les opérateurs du type elliptique" Bull. Sci. Math. , 56 (1932) pp. 316–352
[3]	G. Giraud, "Nouvelle méthode pour traité certains problèmes relatifs aux équations du type elliptique" J. Math. Pures. Appl. , 18 (1939) pp. 111–143
[4]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)