ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS EXERCISES I (HARMONIC FUNCTIONS) MATANIA BEN-ARTZI. BOOKS [CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Interscience Publ. 962. II, [E] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics Vol. 9, AMS 998. [G] P. R. Garabedian, Partial Differential Equations, Second Edition, Chelsea Publ. 986. [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag 977. [J]F. John, Partial Differential Equations, Fourth Edition, Springer-Verlag 986. [PW] M.H.Protter and H.F.Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Publ. 967. [W] H.F.Weinberger, A First Course in Partial Differential Equations, Wiley Publ. 965. ****************************************************************************************** Notation R n A bounded open domain with smooth boundary. G Green s function of. B R (y) The open ball (in R n ) of radius R, centered at y. w ν The outward normal derivative of w at. ds The (Lebesgue) surface measure on. ω n The volume (Lebesgue measure) of B (0). nω n The surface (Lebesgue) measure of B (0). x The Euclidean norm in R n. { Γ(x) = n(2 n)ω n x 2 n if n > 2, 2π if n = 2.
2 MATANIA BEN-ARTZI ***************************************************************************************** () (a) Prove that in spherical coordinates (defined by (x, y, z) = (r cosϕ sinθ, r sinϕ cosθ, r cosθ) ) The Laplacian is given by, { u = r 2 sinθ r (r2 sinθ u r ) + u (sinθ θ θ ) + ϕ ( u } sinθ ϕ ) (b) Show that, up to an additive constant, a radially symmetric harmonic function is given by c Γ(x). (2) Describe the solution of Dirichlet s problem in a rectangle and in a cube by means of separation of variables. (see [W], Ch. IV, Sec. 23, p. 95 and Ch. VI, Sec. 32, p. 46). In particular, construct Green s function for a rectangle. (see [W], Ch. V, Sec.30, p.37). (3) Let u be harmonic in the open disk (i.e., n = 2) B R (0) and continuous in its closure. Let f(θ) be its boundary value (polar coordinates r, θ). Prove Poisson s formula u(r, θ) = R2 r 2 2 π π π f(φ) dφ r 2 + R 2 2 R r cos(θ φ), valid for r < R. Compare this result with the derivation by Fourier series, using separation of variables (see [W], Ch. IV, Sec. 24). (4) (Poisson s formula in a half-plane). Let D = {(x, y) R 2, y > 0} and let g C 0 (R) L (R). Define in D the function f(x, y) by f(x, y) = π yg(t) (t x) 2 dt, (x, y) D. + y2 Prove that: (a) f(x, y) is harmonic in D and can be extended continuously to D, so that f(x, 0) = g(x) for x R. (b) sup{f(x, y), (x, y) D} = sup{g(x), x R}, inf{f(x, y), (x, y) D} = inf{g(x), x R}. (5) Let u = f in. Show that the Kelvin transform of u, defined by satisfies v(x) = x 2 n u(x/ x 2 ) for x/ x 2 v(x) = x n 2 f(x/ x 2 ). In particular, note that v is harmonic (where?) if u is harmonic. (see [GT], Problem 4.7, p. 67). (6) Prove the following stronger form of the Mean Value Property : Suppose that the Dirichlet problem can be solved in for every continuous (given) boundary function. Let u be continuous in and assume that for every interior point x there exists η > 0 such that B η (x) and u(x)
ELLIPTIC PDE EXERCISES 3 is equal to the mean value of u on B η (x). Then u is harmonic in. (see [CH], Ch. IV, Sec. 3, p.279). (7) (Schwarz reflection principle). Let u(x, x 2,..., x n ) be continuous in the closed half-space x 0 and harmonic in its interior x > 0. Assume further that u vanishes on the boundary x = 0. Extend u to the lower half-space by u(x, x 2,..., x n ) = u( x, x 2,..., x n ) for x < 0. Prove that the extended function is harmonic in the whole space R n. Generalize to the case that u is harmonic only in a subdomain (with part of its boundary in x = 0) of the upper half-space. (see [GT], Problem 2.4, p. 28). (8) (Harnack s Inequality) Let u(x) 0 be harmonic in B R (0) and continuous in the closure of the ball. Prove that, for every y B R (0), R n 2 (R y ) (R + y ) n u(0) u(y) Rn 2 (R + y ) u(0). (R y ) n (see [J],Sec. 4.3, p. ). (9) (a) (Liouville s Theorem) Prove that a harmonic function defined in R n and bounded above is constant. (see [GT], Problem 2.4, p. 29). (b) Prove that a function u which is harmonic in the open half-space x > 0, continuous in its closure and vanishes on the boundary x = 0, is identically zero if in addition it satisfies the growth condition u(x) = O( x α ) as x for some exponent α <. (0) (Uniqueness by the energy method) (a) Let u C 2 () C (). Prove the identity u udx = u u ν ds (b) Let u be harmonic in and assume that u 2 dx. u u = 0 on Γ, ν = 0 on Γ 2, where Γ Γ2 =, and Γ has positive (ds) measure. Prove that u 0. (see [W], Ch. III, Sec., p. 52). () (Dirichlet s Principle) For every u C 2 () C () define the Dirichlet form D(u) by D(u) = u 2 dx. Let ϕ C 0 ( ) and let A C 2 () C () consist of all u such that u = ϕ on. (a) Prove that if u A is harmonic then ( ) D(u) D(v), for all v A. (b) Conversely, prove that if u A satisfies (*), then it is harmonic. (see [E], Sec. 2.2, p.42).
4 MATANIA BEN-ARTZI (2) Let u C 2 () C 0 () satisfy the equation u u 2 = 0. Show that u cannot attain its maximum at an interior point unless u 0. (3) Let u C 2 () C 0 () satisfy the equation u u 3 = 0. Show that if u 0 on the boundary then u 0 in. (4) Consider the function u(x, y) = (x2 +y 2 ) ( x) 2 +y 2. Show that it is harmonic and positive in the unit disk {x 2 + y 2 < } and vanishes on its boundary except for one point. Is it a contradiction to the maximum principle? (5) We assume here that R 2. Let u be harmonic in and continuous in except possibly at a point (x 0, y 0 ). Let R > 0 be such that the disk of radius R centered at (x 0, y 0 ) contains. Suppose that: (i) u M on \ (x 0, y 0 ). (ii) 2R 2 u(x, y)/log (x x 0 ) 2 + (y y 0 ) 2 0, as (x, y) (x 0, y 0 ). Show that u M in. (See [W], Ch. IV, Sec. 25, p.2). (6) (a) (The Hadamard Three-Circle Theorem) Here R 2 is an annular domain given by = {(x, y), 0 < a 2 < x 2 + y 2 < b 2, }. Let u C 2 () be subharmonic, i.e., u 0 and denote M(r) = max u(x, y), a < r < b. x 2 +y 2 =r2 Prove that M(r) is a convex function of log r, i.e., that for a < r < r < r 2 < b, M(r) M(r )log(r 2 /r) + M(r 2 )log(r/r ). log(r 2 /r ) (b) (The Hadamard Three-Sphere Theorem) Here R n, spherical shell between concentric spheres given by = {x R n, 0 < a < x < b, }. Let u C 2 () be subharmonic, i.e., u 0 and denote M(r) = max u(x), a < r < b. x =r Prove that for a < r < r < r 2 < b, M(r) M(r )(r 2 n r2 2 n ) + M(r 2 )(r 2 n r 2 n ) r 2 n r2 2 n. n 3, is a (c) (Liouville s Theorem stronger version in R 2 ). Let u C 2 (R 2 \ (0, 0)) be subharmonic and uniformly bounded above. Then u is a constant. (d) Consider the radial function u in R 3 given by, { u(r) = 8 (5 0r2 + 3r 4 ), for r, r, for r. Show that it is subharmonic in all of R 3 and is uniformly bounded. Thus the theorem of the previous part cannot be generalized to higher
ELLIPTIC PDE EXERCISES 5 dimensions. (See [PW], Ch. 2, Sec. 2, p. 28). (7) Show that if u is harmonic in then it is analytic there. (see [E], Sec. 2.2, p.3). Institute of Mathematics, Hebrew University, Jerusalem 9904, Israel E-mail address: mbartzi@math.huji.ac.il