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.