I am interested in parabolic (initial-)boundary value problems violating the so called "complementing condition" (as defined in e.g. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralceva, MR 241822 Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 .)

A model problem: Take the domain to be the half space $\mathbb{R}^2_+ = \{x_1,x_2 \in \mathbb{R}: x_1>0\}$ with boundary $\{x_1 = 0\}$ and consider the heat equation for $u$ given by

$\partial_t u - \triangle u = 0$

$ \triangle u = g$ for $x_1=0$

$u_{|t=0} = 0$

where $g$ is some boundary data (say in some Holder space).

Assuming everything is sufficiently smooth, one can find that this is equivalent to solving the homogeneous, zero initial value heat equation for $u$ with Dirichlet boundary condition

$u(0,x_2,t) = \int_0^t g(x_2,s) ds$.

The $u$ obtained this way solves the above problem - however, it does not satisfy the usual parabolic regularity estimates: one would expect that a second order boundary condition leads to gaining 2 derivatives in space and one derivative in time, however one only gains a time derivative with respect to $g$, i.e. a "loss of derivative" appears.

My issue is: if one perturbs the heat operator (i.e. considers $\partial_t - \triangle u + Eu$ where $E$ is some second order differential operator, "small" in some appropriate sense), can one still solve this equation? The "usual" method (freezing coefficients and solving a fixed point problem) does not seem to work, as a consequence of the loss of derivatives.

The desired result would be that the perturbed problem still allows a solution, and one has a regularity estimate (something like $|u|_{l,\frac{l}{2}} \leq C |g|_{l,\frac{l}{2}}$ with e.g. $||_{l,\frac{l}{2}}$ the parabolic Holder norm of order $l$).

Any references to similar problems would also be very helpful.


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