The Double Layer Potential Operator Through Functional Calculus

Sammanfattning: Layer potential operators associated to elliptic partial differential equations have been an object of investigation for more than a century, due to their contribution in the solution of boundary value problems through integral equations.In this Licentiate thesis we prove the boundedness of the double layer potential operator on the Hilbert space of square integrable functions on the boundary, associated to second order uniformly elliptic equations in divergence form in the upper half-space, with real, possibly non-symmetric, bounded measurable coefficients, that do not depend on the variable transversal to the boundary. This uses functional calculus of bisectorial operators and is done through a series of four steps.The first step consists of reformulating the second order partial differential equation as an equivalent first order vector-valued ordinary differential equation in the upper halfspace. This ordinary differential equation has a particularly simple form and it is here that the bisectorial operator corresponding to the original divergence form equation appears as an infinitesimal generator.Solving this ordinary differential through functional calculus comprises the second step. This is done with the help of the holomorphic semigroup associated to the restriction of the bisectorial operator to an appropriate spectral subspace; the restriction of the operator is a sectorial operator and the holomorphic semigroup is well-defined on the spectral subspace.The third step is the construction of the fundamental solution to the original divergence form equation. The behaviour of this fundamental solution is analogous to the behaviour of the fundamental solution to the classical Laplace equation and its conormal gradient of the adjoint fundamental solution is used as the kernel of the double layer potential operator. This third step is of a different nature than the others, insofar as it does not involve tools from functional calculus.In the last step Green’s formula for solutions of the divergence form partial differential equation is used to give a concrete integral representation of the solutions to the divergence form equation. Identifying this Green’s formula with the abstract formula derived by functional calculus yields the sought-after boundedness of the double layer potential operator, for coefficients of the particular form mentioned above.