Hopf lemma

In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained.

In the special case of the Laplacian, the Hopf lemma had been discovered by Stanisław Zaremba in 1910. ^[1] In the more general setting for elliptic equations, it was found independently by Hopf and Olga Oleinik in 1952, although Oleinik's work is not as widely known as Hopf's in Western countries. ^{[2] [3]} There are also extensions which allow domains with corners. ^[4]

Statement for harmonic functions

Let Ω be a bounded domain in Rⁿ with smooth boundary. Let f be a real-valued function continuous on the closure of Ω and harmonic on Ω. If x is a boundary point such that f(x) > f(y) for all y in Ω sufficiently close to x, then the (one-sided) directional derivative of f in the direction of the outward pointing normal to the boundary at x is strictly positive.

Proof for harmonic functions

Subtracting a constant, it can be assumed that f(x) = 0 and f is strictly negative at interior points near x. Since the boundary of Ω is smooth there is a small ball contained in Ω the closure of which is tangent to the boundary at x and intersects the boundary only at x. It is then sufficient to check the result with Ω replaced by this ball. Scaling and translating, it is enough to check the result for the unit ball in Rⁿ, assuming f(x) is zero for some unit vector x and f(y) < 0 if |y| < 1.

By Harnack's inequality applied to −f

${\displaystyle \displaystyle {-f(rx)\geq -{1-r \over (1+r)^{n-1}}f(0),}}$

for r < 1. Hence

${\displaystyle \displaystyle {{f(x)-f(rx) \over 1-r}={-f(rx) \over 1-r}\geq -{1 \over (1+r)^{n-1}}f(0)>-{f(0) \over 2^{n-1}}>0.}}$

Hence the directional derivative at x is bounded below by the strictly positive constant on the right hand side.

General discussion

Consider a second order, uniformly elliptic operator of the form

${\displaystyle Lu=a_{ij}(x){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+b_{i}(x){\frac {\partial u}{\partial x_{i}}}+c(x)u,\qquad x\in \Omega .}$

In particular, the smallest eigenvalue of the real symmetric matrix ${\displaystyle a_{ij}(x)}$ is bounded from below by a positive constant that is independent of ${\displaystyle x}$. Here ${\displaystyle \Omega }$ is an open, bounded subset of ${\displaystyle \mathbb {R} ^{n}}$ and one assumes that ${\displaystyle c\leq 0}$.

The Weak Maximum Principle states that a solution of the equation ${\displaystyle Lu=0}$ in ${\displaystyle \Omega }$ attains its maximum value on the closure ${\displaystyle {\overline {\Omega }}}$ at some point on the boundary ${\displaystyle \partial \Omega }$. Let ${\displaystyle x_{0}\in \partial \Omega }$ be such a point, then necessarily

${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})\geq 0,}$

where ${\displaystyle \partial /\partial \nu }$ denotes the outer normal derivative. This is simply a consequence of the fact that ${\displaystyle u(x)}$ must be nondecreasing as ${\displaystyle x}$ approach ${\displaystyle x_{0}}$. The Hopf Lemma strengthens this observation by proving that, under mild assumptions on ${\displaystyle \Omega }$ and ${\displaystyle L}$, we have

${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0.}$

A precise statement of the Lemma is as follows. Suppose that ${\displaystyle \Omega }$ is a bounded region in ${\displaystyle \mathbb {R} ^{2}}$ and let ${\displaystyle L}$ be the operator described above. Let ${\displaystyle u}$ be of class ${\displaystyle C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}$ and satisfy the differential inequality

${\displaystyle Lu\geq 0,\qquad {\textrm {in}}~\Omega .}$

Let ${\displaystyle x_{0}\in \partial \Omega }$ be given so that ${\displaystyle 0\leq u(x_{0})=\max _{x\in {\overline {\Omega }}}u(x)}$. If (i) ${\displaystyle \Omega }$ is ${\displaystyle C^{2}}$ at ${\displaystyle x_{0}}$, and (ii) ${\displaystyle c\leq 0}$, then either ${\displaystyle u}$ is a constant, or ${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0}$, where ${\displaystyle \nu }$ is the outward pointing unit normal, as above.

The above result can be generalized in several respects. The regularity assumption on ${\displaystyle \Omega }$ can be replaced with an interior ball condition: the lemma holds provided that there exists an open ball ${\displaystyle B\subset \Omega }$ with ${\displaystyle x_{0}\in \partial B}$. It is also possible to consider functions ${\displaystyle c}$ that take positive values, provided that ${\displaystyle u(x_{0})=0}$. For the proof and other discussion, see the references below.

See also

• Hopf maximum principle

References

1. M.S. Zaremba, Sur un problème mixte relatif à l’équation de Laplace, Bull. Intern. de l’Acad. Sci. de Cracovie, Ser. A, Sci. Math. (1910), 313–344.
2. Hopf, Eberhard. A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3 (1952), 791–793.
3. Oleĭnik, O. A. On properties of solutions of certain boundary problems for equations of elliptic type. Mat. Sbornik N.S. 30 (1952), no. 72, 695–702.
4. Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209–243.

• Evans, Lawrence (2000), Partial Differential Equations, American Mathematical Society, ISBN   0-8218-0772-2
• Fraenkel, L. E. (2000), An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, ISBN   978-0-521-461955
• Krantz, Steven G. (2005), Geometric Function Theory: Explorations in Complex Analysis, Springer, pp. 127–128, ISBN   0817643397
• Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (2nd ed.), Springer, ISBN   9781441970541 (The Hopf lemma is referred to as "Zaremba's principle" by Taylor.)

External links

• Hayk Mikayelyan, Henrik Shahgholian Hopf's lemma for a class of singular/degenerate PDE-s
• Hopf's lemma for a class of fractional singular/degenerate PDE-s
• D. E. Apushkinskaya, A. I. Nazarov A counterexample to the Hopf-Oleinik lemma (elliptic case)