We show that if u is a solution to a linear elliptic differential equation of order 2m > 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space Lp(Rn) or Sobolev space W ± p 1(Rn). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.
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Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values. 2010 Mathematics Subject Classification: 35J67, 35J30, 31B10. (*) Steve Hofmann is partially supported by the NSF grant DMS-1664047. Svitlana Mayboroda is partially supported by the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, the NSF Materials Research Science and Engineering Center Seed Grant, and the Simons Fellowship.
- Dirichlet boundary values
- Elliptic equation
- Higher order differential equation
- Neumann boundary values