Estimates for the L^q-mixed problem in C^1,1-domains

ABSTRACT

We consider the $L^q$-mixed problem in domains in ${\mathbf{\text{R}}}^n$ with $C^{1,1}$-boundary. We assume that the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz. With these assumptions, we show that we may solve the $L^q$-mixed problem for q in the range $1<q<n/(n-1)$.

Keywords:

• Mixed boundary value problem
• Zaremba's problem
• Besov spaces
• Laplace's equation
• harmonic analysis

AMS SUBJECT CLASSIFICATION:

• 35J25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

R. M. Brown is partially supported by the Simons Foundation, Collaboration Grants for Mathematicians, grant number #422756.