The Mixed Boundary Value Problem in Lipschitz Domains

Transcription

1 The Mixed Boundary Value Problem in Lipschitz Domains Katharine Ott University of Kentucky Women and Mathematics Institute for Advanced Study June 9, 2009 Katharine Ott (UK) Mixed Problem 06/09/ / 18

2 Classical Boundary Value Problems for the Laplacian Dirichlet Problem: (D) u C 2 (Ω), u = 0 in Ω, u Ω = f L p ( Ω). Neumann Problem: (N) u C 2 (Ω), u = 0 in Ω, u ν Ω = f L p 0 ( Ω), where ν denotes the outward unit normal vector. Katharine Ott (UK) Mixed Problem 06/09/ / 18

3 Function Spaces Definition. L p ( Ω), 1 < p < is the Lebesgue space of p-integrable functions on Ω, { ( } 1/p L p ( Ω) := f : f dσ) p < +, Ω where dσ denotes surface measure on Ω. Further, define and L p 0 ( Ω) := { f L p ( Ω); Ω fdσ = 0}, L p 1 ( Ω) := {f Lp ( Ω); τ f L p ( Ω)}. Katharine Ott (UK) Mixed Problem 06/09/ / 18

4 Lipschitz Domains A function φ : R n R is Lipschitz if there exists a constant M > 0 such that for any x, y in the domain of φ, φ(x) φ(y) < M x y. Ω is a Lipschitz domain if Ω locally given by the graph of a Lipschitz function φ. Katharine Ott (UK) Mixed Problem 06/09/ / 18

5 History B. Dahlberg [1977,1979], E. Fabes, M. Jodeit, N. Riviere [1978]: (D) is well-posed p (1, ) in the class of smooth domains. B. Dahlberg [1977,1979]: (D) is well-posed p [2, ) in the class of Lipschitz domains. This range is sharp. B. Dahlberg, C. Kenig [1987]: (N) is well-posed p (1, 2] in the class of Lipschitz domains. This range is sharp. C. Kenig [1984]: Counterexamples. Katharine Ott (UK) Mixed Problem 06/09/ / 18

6 The Mixed Problem for the Laplacian Let Ω R n be a bounded open set. Split the boundary of the domain Ω into a Dirichlet and Neumann portion so that Ω = D N, D Ω and N = Ω \ D. Assume D Ω is relatively open, denote by Λ the boundary of D (with respect of Ω). (MP) u = 0 u = f D u ν = f N in Ω on D on N where, as before, ν denotes the outward unit normal vector on Ω. Katharine Ott (UK) Mixed Problem 06/09/ / 18

7 Motivation for Studying (MP) Example 1: Iceberg Consider an iceberg Ω partially submerged in water. Solution to (MP), u(x), is the temperature at each point x Ω. D is the portion of Ω underneath the waterline. Here, Ω behaves like a thermostat so Dirichlet boundary conditions are imposed. N is the portion of Ω above the waterline. Here, Ω acts like an insulator so Neumann boundary conditions are imposed. Katharine Ott (UK) Mixed Problem 06/09/ / 18

8 Motivation for Studying (MP) Example 2: Metallurgical Melting Ω is the cross section of an infinitely long solid with thermal sources located within. u(x) is the temperature of the solid at each point x Ω. Ω = Γ 1 Γ 2. On Γ 1, u is cooled to 0 by a distribution of heat sinks. On Γ 2, the heat u is leaving through Γ 2 at a steady rate g. Mathematical model takes the form u = ρ in Ω u Γ1 = 0 u ν Γ 2 = g Above, ρ is a source function capturing the input of energy into Ω. Katharine Ott (UK) Mixed Problem 06/09/ / 18

9 The Mixed Problem for the Laplacian (MP) u = 0 u = f D u ν = f N in Ω on D on N Goal. Given than f D is in a certain function space on D, and u ν = f N is in a certain function space on N, deduce information about u on the whole boundary Ω. Via trace theorems obtain results for u on Ω. Katharine Ott (UK) Mixed Problem 06/09/ / 18

10 An Example Expectation. (MP) u = h in Ω u D = f D L 2 1 ( Ω) u ν N = f N L 2 ( Ω) u L 2 ( Ω). In the setting of (MP), our intuition that a smooth boundary is better does not hold. Counterexample. Let Ω R 2, Ω := {(x, y) : x 2 + y 2 < 1, y > 0}. Take u(x, y) = Im (x + iy) 1/2. In polar coordinates, u(x, y) = U(r, θ) = r 1/2 sin(θ/2). Katharine Ott (UK) Mixed Problem 06/09/ / 18

11 An Example, continued u(x, y) = U(r, θ) = r 1/2 sin(θ/2) Calculus: Then u = 0 in Ω. u(x, y) = 2 u x u y 2, More calculus: u ν = U θ 1 r, so = 2 U r r U r U. r 2 θ 2 u ν N = r 1/2 cos( θ 2 ) 1 2 N = 0. Katharine Ott (UK) Mixed Problem 06/09/ / 18

12 An Example, continued u(x, y) = U(r, θ) = r 1/2 sin(θ/2) u = 0 in Ω u D = 0 u ν N = 0 u satisfies u D L 2 u 1 (D), ν N L 2 (N). However. u r 1/2 which is not in L 2 ( Ω). Problem is at the origin. Katharine Ott (UK) Mixed Problem 06/09/ / 18

13 More General Example u(x, y) = U(r, θ) = r β sin(βθ). Then u = 0 Ω and u D = 0. u ν N = ( U θ 1 r ) N, so u ν N = r β 1 cos(βα)β. In order for u ν N = 0, need βα = π 2 β = π 2α. Further, u r β 1 = r π 2α 1, so u L 2 ( Ω) whenever 1)2 > 1. In other words, when π > α. ( π 2α Leads to the study of (MP) is creased domains. Katharine Ott (UK) Mixed Problem 06/09/ / 18

14 Some History of (MP) Sevare [1997]: Ω a smooth domain, then solution u of (MP) lies in the Besov space B 3/2,2 (Ω). Brown [1994]: Ω a creased domain, f D L 2 1 ( Ω), f N L 2 ( Ω), then there exists a unique solution u with ( u) L 2 ( Ω). Results extended with J. Sykes [1999] to L p ( Ω), 1 < p < 2. Brown, Capgona and Lanzani [2008]: Ω a Lipschitz graph domain in two dimensions with Lipschitz constant M < 1, solutions in L p ( Ω) for 1 < p < p 0 with p 0 = p 0 (M) > 1. Venouziou and Verchota [2008]: L 2 ( Ω) results for (MP) for certain polyhedra in R 3. Katharine Ott (UK) Mixed Problem 06/09/ / 18

15 The Mixed Problem with Atomic Data Let Ω R n be a bounded Lipschitz domain. (MP a ) u = 0 in Ω, u = 0 on D, u ν = a atom for N. a is an atom for Ω if: suppa r (x) for some x Ω, where r (x) = B r (x) Ω, a 1/σ( r (x)), adσ = 0. Ω a is an atom for N if a is the restriction to N of a function a which is an atom for Ω. H 1 (N) is the collection of functions f which can be represented as Σ j λ j a j, where each a j is an atom for N and Σ j λ j <. Katharine Ott (UK) Mixed Problem 06/09/ / 18

16 The Fundamental Estimate Recall r (x) = B r (x) Ω and let Σ k = 2 k r (x) \ 2 k 1 r (x). Theorem 1, R. Brown, KO Let u be a weak solution of (MP a ) with data f N = a an atom for N which is supported in r (x) and f D = 0. There exists q > 1 such that the following estimates hold ( 1/q u dσ) q Cσ( 8r (x)) 1/q, r (x) ( ) 1/q u q dσ Σ k C2 αk σ(σ k ) 1/q, k 3. Here, C, q and α depend only on the Lipschitz character of Ω. Katharine Ott (UK) Mixed Problem 06/09/ / 18

17 L 1 ( Ω) Estimates for (MP) Theorem 2, R. Brown, KO Let u be a weak solution of the mixed problem with f D = 0 and f N = a, where a is an atom for the Hardy space H 1 (N). Then u satisfies ( u) L 1 ( Ω) C. Theorem 3, R. Brown, KO Let u be a weak solution of (MP) with f D H1 1(D) and f N H 1 (N). Then u satisfies ( ) ( u) L 1 ( Ω) C f D H 1 1 (D) + f N H 1 (N). Katharine Ott (UK) Mixed Problem 06/09/ / 18

18 Results for (MP) in Other Function Spaces Goal. Extend Theorem 3 to L p ( Ω), p [1, 1 + ɛ). That is, wish to prove an estimate of the form ( ) ( u) L p ( Ω) C f D L p 1 (D) + f N L p (N). Katharine Ott (UK) Mixed Problem 06/09/ / 18