An Irregular Complex-Valued Solution to a Scalar Linear Parabolic Equation
Jens Frehse, Joanna Meinel · International Mathematics Research Notices · 2008
We show that Nash's celebrated regularity result [6, Continuity of solutions of parabolic and elliptic equations, American Journal of Mathematics, 1958] does not hold for partial differential operators with complex-valued coefficients. In detail, we present a linear scalar parabolic equation on , n ⩾ 3, with complex-valued measurable coefficients whose solution ∈L∞(L2)∩L2(H1) is discontinuous or may be unbounded.