Mathematical Sciences: Higher Order Operators in Non-Smooth Domains

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Work on this project focuses on two areas related to the study of solutions of elliptic partial differential equations. These are the analysis of the biharmonic and polyharmonic equations on non-smooth domains and boundary value problems for second order elliptic divergence form operators with non-smooth coefficients. In past work it was discovered that the range of solvability of higher order elliptic operators on non-smooth domains depended on the dimension of the underlying domain. There remains a gap in the range in which the solution will have a finite norm (above the quadratic). If the domain has dimension four or greater then solutions exist with unbounded p-norms for some value of p greater than two. But the first value of p is not known, nor is it known whether this gap reduces to zero as the dimension increases. Work will be done in an effort to obtain sharp estimates on this range. There is a close connection between the study of differential operators on non-smooth domains and those with non-smooth coefficients. Research will concentrate on establishing representations of solutions through integrals of boundary values. This is a particularly delicate matter. In the case of smooth coefficient operators, the integrals are Lebesgue integrals (the harmonic measure is absolutely continuous with respect to surface measure). This is not necessarily the case in general. Work will be done in establishing criteria which will still guarantee the integral representation through approximation procedures. Once these are established, one can then begin to study the behavior of solutions as the variable is made to approach the domain boundary.