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A perturbation result for prescribing mean curvature. (English) Zbl 0893.35033

The authors consider the problem of finding a conformal metric on the unit ball \(B \subset \mathbb{R}^n\) with vanishing scalar curvature in \(B\) and prescribed mean curvature \(h\) on the boundary \(S = \partial B\). This problem leads to the nonlinear boundary value problem \[ - \Delta = 0, \;u > 0 \quad \text{in }B, \qquad {{\partial u}\over{\partial \nu}} + {{n-1}\over 2} u = {{n-1}\over 2} hu^{(n+1)/(n-1)} \quad \text{on }S. \tag{1} \] To find a solution of this problem the authors look for minimizers of the functional \[ Q_h(u) = {{\int_B | \nabla u | ^2 + ((n-1)/2) \int_S u^2 }\over{((n-1)/2) [\int_S h u^{2n/(n-1)}]^{(n-1)/n} }} \] restricted to the symmetric class \(\Sigma = \{ u \in H^1 \mid \int_S u^{2n/(n-1)} \cdot x = 0 \}\). For \(h = 1\) this functional is closely related to the best constant in the Sobolev trace inequality [this case was studied by J. Escobar in Commun. Pure Appl. Math. 43, No. 7, 857-883 (1990; Zbl 0713.53024)]. The authors indroduce a family of conformal transformations \(\varphi_{p,t}\) \(((p,t) \in B) \) of \(S\) and define the function \(G : B \to \mathbb{R}^n \) by \( G(p,t) = \int_S h \circ \varphi_{p,t} x \). Their main result is the following:
If \(h\) is positive, smooth, close to 1, and satisfies a certain nondegeneracy condition, then if \(\deg(G,B,0) \neq 0 \), there exists a solution of (1).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0713.53024
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