## Critical points of solutions of elliptic equations in two variables.(English)Zbl 0649.35026

We consider solutions u to the uniformly elliptic equation $\sum^{2}_{i,j=1}a_{ij}u_{x_ ix_ j}+\sum^{2}_{i=1}b_ iu_{x_ i}=0$ in a bounded simply connected plane domain $$\Omega$$. We analyse the problem of obtaining information on the critical points (zeros of the gradient) of u from information on the Dirichlet data $$u=g$$ on $$\partial \Omega.$$
Assuming that the coefficients $$a_{ij},b_ i$$ are sufficiently smooth, we prove the following.
1) If the set of points of relative maximum of the Dirichlet data g on $$\partial \Omega$$ has N connected components, then the interior critical points of u are finite in number and the sum of their multiplicities is at most N-1.
2) If g satisfies the above hypothesis and g and $$\partial \Omega$$ are sufficiently smooth, then, on any subdomain $$G\subset \subset \Omega$$ we have $$| \text{grad} u| \geq C\prod^{K}_{i=1}| x-x_ i|^{m_ i}$$ where $$x_ 1,...,x_ K$$; $$m_ 1,...,m_ K$$ are the interior critical points of u and their respective multiplicities.
The problem is also treated when no smoothness assumption is made on the coefficients.
Reviewer: G.Alessandrini

### MSC:

 35J15 Second-order elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

### Keywords:

uniformly elliptic; critical points; Dirichlet data
Full Text:

### References:

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