Parabolic equations with low regularity

Abstract: In this work we study a variational method for treating parabolic equations that yields new results for non-linear equations with low regularity on source and boundary data. We treat mainly strongly parabolic quasilinear equations and systems in divergence form. The basic idea is to compose the parabolic operator with a weighted sum of the identity operator and the Hilbert transformation in the time direction, and in this way obtain a coercive operator. We work with functions having space derivatives in some Lp-space and half order time derivatives in L2. A key to our results is the celebrated theorem by Marcel Riesz concerning the boundedness of the Hilbert transformation on Lp-spaces when p is strictly greater than one.