For dimensions n ≤ 4, any volume m > 0, and any admissible periodic coefficients, an overdetermined semilinear boundary problem admits a positive smooth solution on a smooth bounded domain.
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Positive solutions to general semilinear overdetermined boundary problems
For dimensions n ≤ 4, any volume m > 0, and any admissible periodic coefficients, an overdetermined semilinear boundary problem admits a positive smooth solution on a smooth bounded domain.