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

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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

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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

 

for r < 1. Hence

 

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

General discussion

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Consider a second order, uniformly elliptic operator of the form

 

In particular, the smallest eigenvalue of the real symmetric matrix   is bounded from below by a positive constant that is independent of  . Here   is an open, bounded subset of   and one assumes that  .

The Weak Maximum Principle states that a solution of the equation   in   attains its maximum value on the closure   at some point on the boundary  . Let   be such a point, then necessarily

 

where   denotes the outer normal derivative. This is simply a consequence of the fact that   must be nondecreasing as   approach  . The Hopf Lemma strengthens this observation by proving that, under mild assumptions on   and  , we have

 

A precise statement of the Lemma is as follows. Suppose that   is a bounded region in   and let   be the operator described above. Let   be of class   and satisfy the differential inequality

 

Let   be given so that  . If (i)   is   at  , and (ii)  , then either   is a constant, or  , where   is the outward pointing unit normal, as above.

The above result can be generalized in several respects. The regularity assumption on   can be replaced with an interior ball condition: the lemma holds provided that there exists an open ball   with  . It is also possible to consider functions   that take positive values, provided that  . For the proof and other discussion, see the references below.

See also

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References

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  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.)
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