Arrieta, José M.

Jimenéz-Casas, Ángela

Rodriguez-Bernal, Anibal

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 11José M. Arrieta; Ángela Jimenéz-Casas; Aníbal Rodríguez-BernalRobin type conditions arising from concentrated potentialsIn: Marek Fila and Angela Handlovičová and Karol Mikula and Milan Medveď and Pavol Quittner and Daniel Ševčovič(eds.): Proceedings of Equadiff 11, International Conference on Differential Equations. Czecho-Slovak series,Bratislava, July 25-29, 2005, [Part 2] Minisymposia and contributed talks. Comenius University Press, Bratislava,2007. Presented in electronic form on the Internet. pp. 157--164.Persistent URL: http://dml.cz/dmlcz/700407Terms of use:© Comenius University, 2007Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digitalsignature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czProceedings of Equadiff-112005, pp. 157–164ISBN 978-80-227-2624-5ROBIN TYPE CONDITIONS ARISING FROMCONCENTRATED POTENTIALS∗JOSÉ M. ARRIETA† , ÁNGELA JIMÉNEZ-CASAS,‡ , AND ANIBAL RODRIGUEZ-BERNAL,§Abstract. We analyze the limit of the solutions of an elliptic problem with zero flux boundarycondition when the potential functions are concentrated in a neighborhood of the boundary and thisneighborhood shrinks to the boundary as a parameter goes to zero.We prove that this family of solutions converges in the sup norm to the solution of an elliptic problemwith Robin type condition.This Robin type conditions for the limiting problem comes from the concentrated potentials aroundthe boundary of the domain.Key words. Robin flux conditions, elliptic boundary problems, concentrating integrals1. Introduction. As it is usually accepted that the environmental influence on areacting media is modelled by a boundary condition in a boundary value problem, it isalso common sense to state that boundary conditions of flux type, must also be equivalentto some external action localized in a thin neighborhood of the boundary.In fact elementary application of finite difference schemes to one dimensional bound-ary value problems, show that this analogy or equivalence must have a sounded mathe-matical foundation.Therefore in this paper we present a rigurous mathematical analysis of this questionfor some elliptic (i.e. time independent) linear problems. In particular we show howlinear reaction and flux terms on the boundary condition can be obtained as a result ofa limiting process from terms distibuted and concentrated near the boundary.2. Concentrating integrals. Let Ω ⊂ RN be a bounded domain with smoothboundary ∂Ω = Γ and outward normal vector ~n(x) for x ∈ Γ.We consider the neighborhood of Γ defined as ωε = {x− s~n(x), x ∈ Γ, s ∈ [0, ε)} ⊂ Ω̄for sufficiently small ε. Then, if we consider Γσ = {x−σ~n(x), x ∈ Γ}, Γ0 = Γ, 0 < ε < ε0the “parallel” interior boundary, we have ωε = ∪0≤σ<εΓσ.Using the standard trace theory and local parameterization of the boundary andpartitions of unity we can prove the following results.Lemma 2.1. Assume that v ∈ W s,p(Ω) ⊂ Lq(Γσ) with s > 1p and s −Np ≥ −(N−1)q i.e.q ≤ p(N−1)N−sp . Then for ε small enough:i) The map σ 7→∫Γσvq is continuous.ii) There exists C > 0 independent of ε such thatsupσ∈[0,ε)‖v‖Lq(Γσ) ≤ C‖v‖W s,p(Ω) (2.1)∗Partially supported by Projects BFM2003-03810 and BFM2003-07749-C05-05 / FISI CICYT, Spain.† Depto. de Matemática Aplicada, U. Complutense de Madrid, (jose.arrieta@mat.ucm.es)‡ Grupo de Dinámica No Lineal,ETSI de (ICAI) U. Pontificia Comillas de Madrid,(ajimenez@rec.upcomillas.es)§Depto. de Matemática Aplicada, U. Complutense de Madrid, (arober@mat.ucm.es)157158 J. M. Arrieta, A. Jimenéz-Casas and A. Rodŕıguez-Bernaliii) ∫ωε|v|q =∫ ε0∫Γσ|v|q dS dσ. (2.2)iv) In particular1ε∫ωε|v|q ≤ c‖v‖qW s,p(Ω) (2.3)andlimε7→01ε∫ωε|v|q =∫Γ|v|qNote that these results assert that for smooth test functions, integrals concentratedclose to the boundary converge to integrals on the boundary.With this we can prove then the following result that, in some sense, allows to passto the limit in concentrating integrals, with nonsmooth integrands.Lemma 2.2. We assume that the functions hε on ωε are such that1ε∫ωε|hε(x)|rdx ≤ C < ∞, for some 1 ≤ r < ∞ (2.4)for some constant C independent of ε. Then there exists a subsequence, still denoted byε, and a function h0 ∈ Lr(Γ) (or h0 ∈M(Γ) if r = 1), such thati) For any continuous function in ωε0 , ϕ, we havelimε7→01ε∫ωεhε(x)ϕ(x) dx =∫Γh0(y)ϕ(y) dσ. (2.5)ii) If uε converge to u weakly in W s,p(Ω) with s > 1p and s−Np ≥ −N−1r′ , thenlimε7→01ε∫ωεhε(x)uε dx =∫Γh0(y)u(y) dσ. (2.6)iii) If uε 7→ u0 in W s,p(Ω) and ϕε 7→ ϕ0 in W σ,q(Ω) with s > 1p and σ >1q ands + σ − Np −Nq > −N−1r′ , we have thatlimε7→01ε∫ωεhε(x)uεϕε(x) dx =∫Γh0(y)u0(y)ϕ(y) dσ. (2.7)iv) In particular if we assume the above the hypothesis, and consider (W σ,q(Ω))′ =W−σ,q′(Ω) such that W σ,q(Ω) ⊂ Lr′(Γ), i.e. σ > 1q and σ −Nq ≥ −N−1r′ . Then,1. Lε = 1εXωεhε is a bounded family in (Wσ,q(Ω))′ := W−σ,q′(Ω).2. The operators Bε converge to the operator B0 in L(W s,p(Ω),W−σ,q′(Ω))with s > 1p and σ >1q and s + σ −Np −Nq > −N−1r′ , where< Bε(u), ϕ >=1ε∫ωεhεuϕ 7−→ < B0(u), ϕ >=∫Γh0uϕ (2.8)for u ∈ W s,p(Ω) and ϕ ∈ W σ,q(Ω).Equadiff-11. Concentrated potentials 159Proof. 1. If ϕ ∈ W s,p(Ω) from Lemma 2.1, we have that|Lε(ϕ)| ≤1ε∫ωε|hε||ϕ| ≤[1ε∫ωε|hε|r] 1r[1ε∫ωε|ϕ|r′] 1r′≤ C‖ϕ‖W s,p(Ω).2. If 1r +1m +1n = 1∣∣∣∣1ε∫ωεhεuϕ∣∣∣∣ ≤ [1ε∫ωε|hε|r] 1r[1ε∫ωε|u|m] 1m[1ε∫ωε|ϕ|n] 1n.If r, m, n are such that s0− Np0 ≥ −N−1m with s0 >1p0and σ0− Nq0 ≥ −N−1n with σ0 >1q0,then from Lemma 2.2 and (2.4)∣∣∣∣1ε∫ωεhεuϕ∣∣∣∣ ≤ C‖u‖W s0,p0 (Ω)‖ϕ‖W σ0,q0 (Ω)and Bε from W s0,p0(Ω) into W−σ0,q′0(Ω)) is uniformly bounded. Hence, fixed u ∈ W s0,p0(Ω)we have Lemma 2.1 that< Bε(u), ϕ >=1ε∫ωεhεuϕ 7→∫Γh0uϕ =< B0(u), ϕ >uniformly for ϕ in compact sets of W σ0,q0(Ω). Hence if q ≥ q0 with σ > 1q and σ −Nq >σ0 − Nq0 then Wσ,q(Ω) ⊂ W σ0,q0(Ω) with compact embedding, and then, in particularBε(u) 7→ B0(u) in W−σ,q′(Ω).Again this implies uniform convergence for u in compact sets of W s0,p0(Ω). Hence ifp ≥ p0 with s > 1p and s −Np ≤ s0 −Np0then W s,p(Ω) ⊂ W s0,p0(Ω) with compactembedding, and then, in particular, we haveBε 7→ B0 in L(W s,p(Ω),W−σ,q′(Ω))which gives the result.3. An elliptic problem with concentrating non homogeneous terms. Withthe previous results, in this section we study the behavior, for small ε, of the solutions ofthe elliptic problem −∆uε + uε =1εXωεfε in Ω∂uε∂n= 0 on Γ(3.1)with a given fε and where Xωε denote the characteristic function of the set ωε. Hencethe effective reaction is concentrated on this set.The goal in this section is to prove that, the family of solutions, uε, converges whenthe parameter ε goes to zero, to a limit function, which is given by the solution of thehomogeneous elliptic problem with nonhomogeneous flux conditions on the boundary:{ −∆u + u = 0 in Ω∂u∂n= f0 on Γ.(3.2)160 J. M. Arrieta, A. Jimenéz-Casas and A. Rodŕıguez-BernalMoreover, we give conditions on fε that guarantee that the convergence is uniformin Ω̄ and in C∞loc(Ω). For this we will assume, that fε satisfies1ε∫ωε|fε(x)|r dx ≤ C < ∞ (3.3)for some constant C independent of ε and there exists a function, f0 ∈ Lr(Γ) such thatfor any smooth function ϕ, we havelimε7→01ε∫ωεfε(x)ϕ(x) dx =∫Γf0(y)ϕ(y) dσ. (3.4)with ε0 > 0 sufficiently small.With the notations above, we can prove firstTheorem 3.1.i) Assume the notations above and the hypotheses (3.3) with r = 1 and (3.4). Thenthe family uε converges to u in W 1,q(Ω) for every q < NN−1 , in Ls(Ω) for everys < NN−2 and almost everywhere, where u is the unique solution of (3.2). More-over the family uε also converges in Cj(K) for every j and K ⊂⊂ Ω a compactset.ii) Assume in addition that, for r > 1 we have1ε∫ωε|fε(x)|r dx ≤ C < ∞for some constant C independent of ε. Then uε is uniformly bounded in W 1+1r ,r(Ω)and thereforeuε 7→ u in Wα,q(Ω)with 1 + 1r −Nr > α−Nq . In particular, if r > N − 1 thenuε 7→ u in Cβ(Ω̄)for some β > 0.Proof. i) Let Fε = 1εXωεfε which is uniformly bounded in L1(Ω). Then, there exists λ0such that if λ ≥ λ0 then the elliptic operator is invertible. Hence, by elliptic regularity wehave a uniform bound of solutions, uε, in W 2−δ,1(Ω) for every δ > 0. Thus, there existsa subsequence which converges weakly to u in W 2−δ,1(Ω), for every δ > 0.On the other hand, again from Sobolev’s embedding, we have that for any q < NN−1we can find δ > 0 such that W 2−δ,1(Ω) ⊂ W 1,q(Ω) with compact inclusion. From this, wecan ensure the subsequence converges strongly in W 1,q(Ω) for every q < NN−1 . Moreover,taking into account that, again from Sobolev’s embeddings, for every s < NN−2 we canfind q < NN−1 such that W1,q(Ω) ⊂ Ls(Ω), then the subsequence converge also stronglyin Ls(Ω) and thus almost everywhere. Also, the traces of uε converge to the trace of u inW 1−1q ,q(Γ).Next, we prove u verifies the homogeneous elliptic problem (3.2). In effect, multiply-ing the equation from (3.1) by ϕ ∈ C∞(Ω) we obtain∫Ω(∇uε∇ϕ + uε)ϕ =1ε∫ωεfεϕ.Equadiff-11. Concentrated potentials 161Now we assume first ϕ ∈ C∞c (Ω) and taking the limit as ε goes to zero, using (3.4), weget ∫Ω∇u∇ϕ + uϕ = 0, ∀ϕ ∈ C∞c (Ω). (3.5)Therefore the limit function satisfies −∆u + u = 0 in Ω. In particular u ∈ C∞loc(Ω).Now, we consider a test function ϕ ∈ C∞(Ω̄) and using again (3.4) together with theconvergence of the traces, we get∫Ω∇u∇ϕ +∫Ωuϕ =∫Γf0ϕ, for all ϕ ∈ C∞(Ω̄). (3.6)Using the equation for u in Ω and integrating by parts, we get now∫Γ∂u∂nϕ =∫Γf0ϕ, ∀ϕ ∈ C∞(Ω̄)where ∂u∂n denote the normal derivative defined by∫Γ∂u∂nϕ =∫Ω∇u∇ϕ−∫Ω(−∆u)ϕfor every ϕ ∈ C∞(Ω̄). Therefore u solves (3.2).Now, since λ > λ0, from the uniqueness of solutions for the limit problem (3.2), wehave that the whole family uε converges to u.Next, to improve the convergence in Ω, given K ⊂⊂ Ω we consider the auxiliarfunction vε = uεϕ where ϕ ∈ C∞c (Ω) is a regular function, such that is equal to 1 in K,and 0 in the set Ω \ Ω′ where Ω′ is a domain such that K ⊂ Ω′ ⊂ Ω.Now observe that considering ε small enough such that ωε ⊂ Ω\Ω′ and thus Fεϕ = 0in Ω. Then, vε, satisfies the elliptic problem{−∆vε + vε = gε in Ωvε = 0 on Γ(3.7)with gε = −2∇uε∇ϕ− uε∆ϕ, which satisfies gε ∈ Lq(Ω). From uniform estimates of thefamily uε in W 1,q(Ω), with q < NN−1 , we obtain ‖gε‖Lq(Ω) ≤ C1 for some constant C1 > 0independent of ε. Then, from (3.7) there exists C2 > 0 such that ‖vε‖W 2,q(Ω)∩W 1,q0 (Ω) ≤ C2.Now, applying a bootstrap argument to vε and taking into account that vε = uε onK we have that, for any j there exists C(j) positive constant independent of ε such that‖uε‖Cj(K) ≤ C(j).Thus, using the Ascoli-Arzela’s theorem, and the uniqueness of u we get the result.ii) From Lemma 2.2 the right hand side in the equation of uε defines a boundedfamily in the Sobolev’s space (W s,p(Ω))′ = W−s,p′(Ω) for s > 1p and s−Np = −N−1r′ .Now, from the elliptic regularity, we have the solution of (3.1) satisfies that uε ∈W 2−s,p′(Ω) and is uniformly bounded in this space.Now note that W 2−s,p′(Ω) ⊂ Wα,q(Ω), with compact embedding, provided2− s− Np′= 1− N − 1r= 1 +1r− Nr> α− Nq.In particular, if r > N − 1 we have 1− N−1r > 0 and then W2−s,p′(Ω) ⊂ Cβ(Ω̄) withcompact embedding, for some β.162 J. M. Arrieta, A. Jimenéz-Casas and A. Rodŕıguez-Bernal4. An elliptic problem with concentrating potentials. We assume now thatwe have concentrating potentials Vε and nonhomogeneous terms fε satisfying1ε∫ωε|Vε(x)|ρ dx ≤ C < ∞ (4.1)1ε∫ωε|fε(x)|r dx ≤ C < ∞ (4.2)for some constant C independent of ε, and there exists functions f0 ∈ Lr(Γ), and V0 ∈Lρ(Γ) (or f0, V0 ∈ M(Γ) if r = ρ = 1), such that for any function ϕ, continuous in ωε0 ,we havelimε7→01ε∫ωεfε(x)ϕ(x) dx =∫Γf0(y)ϕ(y) dσ. (4.3)limε7→01ε∫ωεVε(x)ϕ(x)dx =∫ΓV0(y)ϕ(y)dσ. (4.4)In this section we prove that for such concentrating potentials and nonhomogeneousterms, we get uε 7→ u in C(Ω̄) where uε satisfies that−∆uε + λuε = 1εXωεVεuε +1εXωεfε + g in Ω∂uε∂n= 0 on Γ(4.5)and u satisfies −∆u + λu = g in Ω∂u∂n= V0u + f0 on Γ.In fact if s, p, σ are as in Lemma 2.2 iii) and if we define the operator Pε ∈L(W s,p(Ω),W−σ,p(Ω)) as< Pε(uε), ϕ >=1ε∫ωεVεuεϕ,then we haveTheorem 4.1. Assume Vε satisfies (4.1) with ρ > N − 1. Then there exists some λ0such that for λ ≥ λ0 the the elliptic operator A0 + λI − Pε in (4.5) is invertible and‖(A0 + λI − Pε)−1‖W−σ,p(Ω) ≤C|λ|(4.6)for some constant C independent of ε.Proof. Writing (4.5) as a perturbation of a fixed elliptic operator i.e.A0uε + λuε = Pεuε + hεfrom Lemma 2.2 we have Pε 7→ P0 in L(W s,p(Ω),W−σ,p(Ω)) with< Pε(uε), ϕ >=1ε∫ωεVεuεϕ 7→∫ΓV0uϕ =< P0(u), ϕ > (4.7)for every u ∈ W s,p(Ω) and ϕ ∈ W σ,p′(Ω).Equadiff-11. Concentrated potentials 163Since ρ > N − 1 we can take s + σ < 2 and we get that A0 + λI − Pε is well definedfrom W 2−σ,p(Ω) into W−σ,p(Ω). In particular for given g ∈ W−σ,p(Ω) the equationA0uε + λuε − Pεuε = g can be written asuε = (A0uε + λI)−1(g + Pεuε) = T ελ(uε).Now the resolvent estimates for A0 imply that the Lipschitz constant of T ελ is boundedby C|λ| , for some constant C independent of ε, and therefore it is a contraction for largeenough λ.With this we getCorollary 4.2. Assume that λ ≥ λ0 and in addition that, for r > 1 we have1ε∫ωε|fε(x)|r dx ≤ C < ∞Then uε is uniformly bounded in W 1+1r ,r(Ω) and thereforeuε 7→ u in Wα,q(Ω) as ε 7→ 0with 1 + 1r − 1 +Nr > α−Nq where u is the unique solution of−∆u + λu = g in Ω∂u∂n= V0u + f0 on Γ(4.8)In particular, if r > N − 1 thenuε 7→ u in Cβ(Ω̄) as ε 7→ 0for some β > 0.The proofs above can be extended along the same lines above for more general ellipticproblems with smooth coefficients.Theorem 4.3. Under the above notations and the hypotheses for Vε, fε and g, we considerthe family uε given by the solutions of the following problems−div(a(x)∇uε) + c(x)uε + λuε = 1εXωεVεuε +1εXωεfε + g in Ωuε = h, on Γ0,a(x)∂uε∂n+ b(x)uε = j on Γ1(4.9)for sufficiently large λ, and smooth coefficients a, c ∈ C1(Ω̄), and b ∈ C1(Γ̄1) with a(x) ≥a0 > 0.Then, given h ∈ C(Γ̄0) and j ∈ Lr(Γ1), we have that uε 7→ u in C(Ω̄) where u is thesolution of −div(a(x)∇u + c(x)u + λu = gu = h, on Γ0,a(x)∂u∂n+ b(x)u = j + V0u + f0 on Γ1(4.10)The key argument for the proof, relies in considering the scale of extrapolation spacesassociated to these operators, as constructed in [1]. In fact it can be shown that this scaledoes not depend on ε and that the concentrating potentials and nonhomogeneous termsremain uniformly bounded in the corresponding dual norms.164 J. M. Arrieta, A. Jimenéz-Casas and A. Rodŕıguez-Bernal5. Singular coefficients. The analysis above fails if the smoothness of the coeffi-cients in (4.9) is removed, since then the scales of spaces in [1] is no longer available.In this case however, by using a different approach we are still able to prove analogousconvergence results as above; in particular, we are still able to prove the convergence inC(Ω̄). See [2] for details.REFERENCES[1] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.In Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), volume 133of Teubner-Texte Math., Teubner, Stuttgart, 1993.[2] J. M. Arrieta, A. Jiménez-Casas and A. Rodŕıguez-Bernal, Flux terms and Robin boundary conditionas limit of reactions and potentials concentrating in the boundary. To appear in Revista MatemáticaIberoamericana[3] L. Bocardo, D. Giachet, J. I. Dı́az and F. Murat, Existence and regularity of renormalized solutionsfor some elliptic problems involving derivatives of nonlinear term, J. Differential Equations 106(2)(2000), 215–237.[4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, ed. New York, 1977.[5] O. Ladyzhenskaya and N. Uralseva, Linear and Quasilinear Elliptic Equations, Academic Press1968.

Robin type conditions arising from concentrated potentials

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Robin type conditions arising from concentrated potentials

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Institute of Mathematics AS CR, v. v. i.