## Estimation of the best constant involving the $$L^2$$ norm of the higher-order Wente problem.(English)Zbl 1095.53007

The Wente problem arose in the study of surfaces with constant mean curvature. In higher dimensions this is the problem of regularity of a solution $$\varphi$$ of the equation $$(-\Delta)^p\varphi= \det\nabla u$$ in $$\mathbb{R}^{2p}$$ with $$\lim_{|x|\to+\infty}\varphi(x)=0$$ and $$u\in W^{1,2p}(\mathbb{R}^{2p},\mathbb{R}^{2p})$$. The first author proved in 2001 that the solution $$\varphi$$ is in $$L^\infty (\mathbb{R}^{2p})$$ and $$\overline\Delta^{k/2}\varphi$$ is in $$L^{2p/k}(\mathbb{R}^{2p})$$ for $$1\leq k\leq p$$, with the following estimates: $$\|\varphi\|_\infty+\|\overline \Delta^{k/2}\varphi\|_{2p/k}\leq C\|\Delta u\|^{2p}_{2p}$$, where $$\|\overline \Delta^{k/2}\varphi\|_{2p/k}$$ is $$\|\Delta^{k/2}\varphi \|_{2p/k}$$ if $$k$$ is even, and is $$\|\nabla(\Delta^{(k-1)/2})\varphi \|_{2p/k}$$ if $$k$$ is odd. Now it is proved that the best constant $$C_p$$ is achieved by a one parameter family of functions $$\overline\varphi$$ and $$\overline u$$ given by $\overline\varphi(x) =\frac{2}{(2p)!(1+cr^2)}, \quad\overline u=\frac{2\sqrt cx}{1+cr^2},$ where $$c>0$$ is some arbitrary positive constant. In the case $$p=2$$ a more explicit expression of the best constant is given.

### MSC:

 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C40 Global submanifolds
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