Boundary value problems for a second-order elliptic partial differential equation system in Euclidean space

Let omega subset of Double-struck capital Rm$$ \Omega \subset {\mathrm{\mathbb{R}}}<^>m $$ be a bounded regular domain, let partial differential x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in Double-struck capital Rm$$ {\mathrm{\mathbb{R}}}<^>m $$, and let Double-struck capital R0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space Double-struck capital R0,m$$ {\mathrm{\mathbb{R}}}<^>{0,m} $$. For k is an element of{1, horizontal ellipsis ,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, Double-struck capital R0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}<^>{(k)} $$ denotes the space of k$$ k $$-vectors in Double-struck capital R0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second-order elliptic system of partial differential equations of the form partial differential x_Fk partial differential x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in omega$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$-vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$-vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to partial differential omega$$ \mathrm{\partial \Omega } $$, ensuring the well-posedness for the problems. Investigation of the spectral properties of the sandwich operator partial differential x_(.) partial differential x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem-solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of Double-struck capital Rm$$ {\mathrm{\mathbb{R}}}<^>m $$.