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Analysis and numerics of partial differential equations
 

Analysis and numerics of partial differential equations

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    Analysis and numerics of partial differential equations Analysis and numerics of partial differential equations Document Transcript

    • Some Aspects of the Research of EnricoMagenes in Partial Differential EquationsGiuseppe GeymonatAbstract The author traces the initial stage of Enrico Magenes’s research, with aparticular emphasis on his work in Partial Differential Equations. The very fruitfulcollaborations with G. Stampacchia and J.-L. Lions are clearly presented.1 The Beginnings in ModenaThe first researches of Enrico Magenes in Partial Differential Equations date to1952, [14, 15] (and in the same year he became professor of Mathematical Analysisat the University of Modena). Their argument is the application to the heat equationof a method that in the Italian School is called “Picone’s Method”. The basic ideaof the method is to transform the boundary value problem into a system of integralequations of Fischer-Riesz type. This idea was introduced by Picone around 1935and then deeply applied by Amerio, Fichera, and many others to elliptic equations.For simplicity, we present the method in the simplest case of a non-homogeneousDirichlet problem in a smooth, bounded domain Ω ⊂ RN : A(u) = f in Ω, γ0 u = g on Γ := ∂Ω (1)where A(u) is a second order linear elliptic operator with smooth coefficients andγ0 u denotes the trace of u on Γ ; for simplicity one can also assume that uniqueness ∂holds true for this problem. Let A be the formal adjoint of A and let ∂ν denote theso-called co-normal derivative (when A = , then A = and ν = n, the outgoingnormal to Γ ). The Green formula states ∂u ∂w A(u)wdx − uA (w)dx = w−u dΓ. (2) Ω Ω Γ ∂ν ∂νHence, if one knows a sequence wn of smooth enough functions, such that A (wn )and γ0 (wn ) both converge, then, thanks to (2), the determination of the vectorG. Geymonat (B)LMS, Laboratoire de Mécanique des Solides UMR 7649, École Polytechnique, Route de Saclay,91128 Palaiseau Cedex, Francee-mail: giuseppe.geymonat@lms.polytechnique.frF. Brezzi et al. (eds.), Analysis and Numerics of Partial Differential Equations, 5Springer INdAM Series 4, DOI 10.1007/978-88-470-2592-9_2,© Springer-Verlag Italia 2013
    • 6 G. Geymonat(u, ∂u ) with u the solution of (1) is reduced to the solution of a system of linear ∂νequations of Fischer-Riesz type. The difficulty was naturally to find such a sequence, to prove its completeness (ina suitable functional space) and hence the space where the corresponding system issolvable and so the problem (1). Following Amerio [1], let the coefficients of A be smoothly extended to a do-main Ω ⊃ Ω and for every fixed R ∈ Ω let F (P , R), as a function of P , be thefundamental solution of A (w) = 0 (moreover, such a fundamental function can bechosen so that, as a function of R, it also satisfies A(u) = 0). Then from (1) and (2)it follows that for every Q ∈ Ω Ω ∂F (x, Q) ∂u(x) 0= u(x) − F (x, Q) dΓ − f (x)F (x, Q)dx. (3) Γ ∂ν ∂ν Ω ∂u(x)This equation gives a necessary compatibility condition between γ0 u and ∂ν .Moreover, if ϕn is a sequence of “good” functions defined in Ω Ω, then one cantake wn (P ) = ΩΩ ϕn (x)F (P , x)dx. Two problems remain: (i) the choice of the sequence ϕn , in order that the procedure can be applied;(ii) the determination of the good classes of data f, g, Ω, and solutions u to which the procedure can be applied.In this context it is also useful to recall that for every P ∈ Ω it holds 2π N/2 ∂F (x, P ) ∂u(x) u(P ) = u(x) − F (x, P ) dΓ − f (x)F (x, P )dx.Γ (N/2) Γ ∂ν ∂ν Ω (4)In order to study the previous problems, one has to study the fine properties ofthe simple and double layer potentials, appearing in (3) and (4). See for instanceFichera’s paper [3], where many properties are studied, and in particular results ofcompleteness are proved. (The modern potential theory studies the fine propertiesof the representation (4) for general Lipschitz domains and in a Lp framework.) Magenes applied the method to the heat operator E(u) = u − ∂u in Ω × (0, T ), ∂twhose formal adjoint is E (u) = u + ∂u ; the Green formula (2) becomes ∂t T T E(u)wdxdt − uE (w)dxdt 0 Ω 0 Ω T ∂u ∂w = w−u dΓ dt 0 Γ ∂n ∂n + u(T )w(T )dx − u(0)w(0)dx. (5) Ω ΩFollowing the approach of Amerio and Fichera, Magenes used the fundamental so-lution of the heat equation, defined by F (x, t; x , t ) = t 1 exp(− 4(t −x ) for t > t −t x −t)and F (x, t; x , t ) = 0 for t ≤ t. He also defined a class of solutions of the heat
    • Research in PDE 7equation E(u) = f , assuming the boundary value in a suitable way and representedby potentials of simple and double layer. These researches were followed [17] by the study of the so-called mixed prob-lem, where the boundary is splitted in two parts: in the first one the boundary condi-tion is of Dirichlet type, and in the other one the datum is the co-normal derivative.This problem was of particular difficulty for the presence of discontinuity in thedata, even in the stationary case: see [16] (where the results are stated for N = 2,although they are valid for arbitrary N ) and has stimulated many researches usingpotential theory not only of Magenes (see e.g. [18, 19]) but also of Fichera, Miranda,Stampacchia, . . .2 The Years in Genoa with G. StampacchiaFrom the historical point of view, these researches show the change of perspectivethat occurred in Italy at that time in the study of these problems with the use of• some first type of trace theorems (e.g. inspired by the results of Cimmino [2]);• the introduction of the concept of weak solution;• the use of general theorems of functional analysis (see e.g. [4]).Under this point of view, the following summary of a conference of Magenes givesa typical account (see [20]). Breve esposizione e raffronto dei più recenti sviluppidella teoria dei problemi al contorno misti per le equazioni alle derivate parzialilineari ellittiche del secondo ordine, soprattutto dal punto di vista di impostazioni“generalizzate” degli stessi (A short presentation and comparison of the most re-cent developments in the theory of mixed boundary value problems for second orderelliptic linear partial differential equations, mainly from the point of view of “gen-eralized” approaches to them). At the end of 1955 Magenes left the University of Modena for the University ofGenoa, where he had G. Stampacchia as colleague. Stampacchia was a very goodfriend of Magenes from their years as students at Scuola Normale, since both whereantifascist. Moreover, Magenes and Stampacchia were well aware of the fundamen-tal change induced by the distribution theory and the Sobolev spaces in the calculusof variations and in the study of partial differential equations, particularly in thestudy of boundary value problems for elliptic equations (see for instance the bibli-ography of [20]). They studied the works of L. Schwartz and its school, and specially the resultson the mixed problem in the Hadamard sense. At the first Réunion des mathémati-ciens d’expression latine, in September 1957, Magenes and Stampacchia met J.-L.Lions. It was the beginning of a friendship, that would never stop. During the Spring1958, J.-L. Lions gave at Genoa a series of talks on the mixed problems [5, 6], andin June 1958 Magenes and Stampacchia completed a long paper [22], that wouldhave a fundamental influence on the Italian researches on elliptic partial differentialequations. Indeed, that paper gives a general presentation of the results obtained upto that moment in France, United States, Sweden, Soviet Union by N. Aronszajn,
    • 8 G. GeymonatF.E. Browder, G. Fichera, K.O. Friedrichs, L. Gårding, O. Ladyzenskaja, J.-L. Li-ons, S.G. Mikhlin, C.B. Morrey Jr., L. Nirenberg, M.I. Visik, . . . . It is worth giving the titles of the four chapters: I General notions, II Boundaryvalue problems for linear elliptic equations, methods with finite “Dirichlet integral,”III Problems of regularization, IV Other approaches to boundary value problems.Then, in the following few years Magenes tried to increase the audience of thismethodology in the Italian mathematical community, giving lectures in various uni-versities (see e.g. [21]), for instance organizing with Stampacchia a CIME courseon distribution theory in 1961, . . . At the end of 1959 Magenes left Genoa and went to the University of Pavia.During the year 1959, the collaboration with J.-L. Lions became more active andthey started a long series of joint works [7–10], whose results were summarizedand fully developed in a series of books [11–13] translated in Russian, English andChinese.3 The Collaboration with J.-L. Lions in the Study of Boundary Value ProblemsFollowing an idea of J. Hadamard, Courant and Hilbert (Methods of MathematicalPhysics, volume II, Interscience, 1962, p. 227) state that a mathematical problem,which must correspond to a physical reality, should satisfy the following basic re-quirements:1. The solution must exist.2. The solution should be uniquely determined.3. The solution should depend continuously on the data (requirement of stability).In order to satisfy these requirements, one has to identify the functional spaceswhere the problems are well-posed. The distribution theory and the Sobolev spacesgive a natural framework and the instruments to study partial differential equations.The results collected in the first three chapters of [22] allow to prove that the el-liptic boundary value problems with homogeneous boundary data are well posedin Sobolev spaces W m,2 (Ω) with m big enough. For non-homogeneous boundarydata, the situation was more difficult, since at first it was necessary to give a gooddefinition of the trace γ0 u of an element u ∈ W m,p (Ω) on Γ := ∂Ω. The gooddefinitions and the corresponding characterizations were given (under various con-ditions on p ≥ 1, on m > 1 − p and on the regularity of the domain Ω, i.e. of 1its boundary Γ ) by E. Gagliardo, J.-L. Lions, and P.I. Lizorkin, G. Prodi, . . . . Inparticular it was proved that the trace operator cannot be continuously defined onL2 (Ω). However, for many problems coming from the applications (e.g. mechanics, en-gineering, . . .) the natural setting is in Sobolev spaces of low order and sometimesof negative order. Therefore, it is necessary to define a weak or generalized solutionof a non-homogeneous boundary value problem and hence, to give a good definitionof trace in a weak sense.
    • Research in PDE 9 Inspired by the theory of distributions, Lions and Magenes [7–10] tackled theproblem by duality. More precisely let us consider the map u → Au := {Au, Bγ0 u},where A is a linear elliptic operator with smooth coefficients defined in a domainΩ ⊂ RN with smooth boundary Γ , and Bγ0 is a linear differential operator withsmooth coefficients, defined on Γ , and compatible with A in a suitable sense. Such a general framework is a “natural” extension of the Dirichlet problem (1).Thanks to known regularity results (described for instance in Chap. III of [22]), themap A : E(Ω) → F (Ω) × G(Γ ) is an isomorphism (for simplicity, and in general afinite index operator) between the Sobolev spaces E(Ω) and F (Ω) × G(Γ ), wherethese spaces are of big enough positive order. In the case of (1), one can take for instance E(Ω) = H m+2 (Ω)(= W m+2,2 (Ω))with m ≥ 0, and then F (Ω) = H m (Ω) and G(Γ ) = H m+3/2 (Γ ). By restriction tothe case of homogeneous boundary data and to the space F0 (Ω) (closure of D(Ω)into F (Ω)), it is possible to define the isomorphism A : X(Ω) → F0 (Ω), whereX(Ω) is a subspace of E(Ω). By transposition, for every linear and continuous form L(v) on X(Ω), thereexists u ∈ (F0 (Ω)) such that u, A (v) = L(v) for all v ∈ X(Ω). (6)Let us point out that (F0 (Ω)) is a Sobolev space of negative order (in the case of(1) F0 (Ω) = H0 (Ω) and (F0 (Ω)) = H −m (Ω)). In order to get the wanted result, mLions and Magenes chose L = L1 + L2 in such a way that L1 gives rise to theequation A u = f , where A is the linear elliptic operator formally adjoint to A,and L2 corresponds to the non-homogeneous boundary conditions B u = g in themost natural way. Perhaps the most interesting contribution of Lions and Magenes was the optimalchoice of L2 . It was obtained thanks to a clever use of the Green formula, that allowsto naturally define the traces of every element u ∈ (F0 (Ω)) , such that A u belongsto a suitable distribution space on Ω. For instance, in the case of (1) with A = A = , Bγ0 = ∂n := γ1 and m = 0, ∂one can define the trace γ0 u ∈ H −1/2 (Γ ) for every u ∈ L2 (Ω), such that A u = u ∈ L2 (Ω). The main steps of the proof are the following:1. One proves the density of D(Ω) into the space Y (Ω) := {u ∈ L2 (Ω); u ∈ L2 (Ω)}, equipped with the natural graph norm.2. Let us define X(Ω) = {v ∈ H 2 (Ω); γ0 u = 0} and let us remark that the map v −→ γ1 v is a linear and continuous map of X(Ω) onto H 1/2 (Γ ), whose kernel 2 is H0 (Ω).3. For every (u, φ) ∈ Y (Ω) × H 1/2 (Γ ), one defines the bilinear and bi-continuous map L2 (u, φ) with L2 (u, φ) = u vφ dx − uvφ dx, Ω Ω where vφ ∈ X(Ω) is such that γ1 vφ = φ (it is easy to verify that indeed L2 (u, φ) = 0 when vφ ∈ H0 (Ω) and hence, L2 (u, φ) does not depend on the 2 particular choice of vφ ).
    • 10 G. Geymonat4. One can do the identification L2 (u, φ) = T u, φ , where •,• denotes the duality pairing between H −1/2 (Γ ) and H 1/2 (Γ ), and u −→ T u is linear and continuous from Y (Ω) to H −1/2 (Γ ).5. When u ∈ D(Ω), then the Green formula (2) implies L2 (u, φ) = uφdΓ Γ and hence, the map T can be identified with the trace map. The books [11] and [12] present the general theory, not only for elliptic operators,but also for linear evolution equations of parabolic type, both in distributions spacesand also [13] in ultra-distributions of Gevrey classes.References 1. Amerio, L.: Sul calcolo delle soluzioni dei problemi al contorno per le equazioni lineari del secondo ordine di tipo ellittico. Am. J. Math. 69, 447–489 (1947) 2. Cimmino, G.: Sulle equazioni lineari alle derivate parziali di tipo ellittico. Rend. Semin. Mat. Fis. Milano 23, 1–23 (1952) 3. Fichera, G.: Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di fun- zioni. Ann. Mat. Pura Appl. (4) 27, 1–28 (1948) 4. Fichera, G.: Methods of functional linear analysis in mathematical physics: “a priori” es- timates for the solutions of boundary value problems. In: Proceedings of the International Congress of Mathematicians, vol. III, Amsterdam, 1954, pp. 216–228. North-Holland, Ams- terdam (1956) 5. Lions, J.-L.: Problemi misti nel senso di Hadamard classici e generalizzati. Rend. Semin. Mat. Fis. Milano 28, 149–188 (1958) 6. Lions, J.-L.: Problemi misti nel senso di Hadamard classici e generalizzati. Rend. Semin. Mat. Fis. Milano 29, 235–239 (1959) 7. Lions, J.-L., Magenes, E.: Problemi ai limiti non omogenei. I. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 14, 269–308 (1960) 8. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes. II. Ann. Inst. Fourier (Grenoble) 11, 137–178 (1961) 9. Lions, J.-L., Magenes, E.: Problemi ai limiti non omogenei. III. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 15, 41–103 (1961)10. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes. VII. Ann. Mat. Pura Appl. (4) 63, 201–224 (1963)11. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968)12. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 2. Dunod, Paris (1968)13. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 3. Dunod, Paris (1970)14. Magenes, E.: Sull’equazione del calore: teoremi di unicità e teoremi di completezza connessi col metodo di integrazione di M. Picone, Nota I. Rend. Semin. Mat. Univ. Padova 21, 99–123 (1952)15. Magenes, E.: Sull’equazione del calore: teoremi di unicità e teoremi di completezza connessi col metodo di integrazione di M. Picone, Nota II. Rend. Semin. Mat. Univ. Padova 21, 136– 170 (1952)
    • Research in PDE 1116. Magenes, E.: Sui problemi al contorno misti per le equazioni lineari del secondo ordine di tipo ellittico. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 8, 93–120 (1954)17. Magenes, E.: Problemi al contorno misti per l’equazione del calore. Rend. Semin. Mat. Univ. Padova 24, 1–28 (1955)18. Magenes, E.: Problema generalizzato di Dirichlet e teoria del potenziale. Rend. Semin. Mat. Univ. Padova 24, 220–229 (1955)19. Magenes, E.: Sulla teoria del potenziale. Rend. Semin. Mat. Univ. Padova 24, 510–522 (1955)20. Magenes, E.: Recenti sviluppi nella teoria dei problemi misti per le equazioni lineari ellittiche. Rend. Semin. Mat. Fis. Milano 27, 75–95 (1957)21. Magenes, E.: Sui problemi al contorno per i sistemi di equazioni differenziali lineari ellittici di ordine qualunque. Univ. Politec. Torino. Rend. Semin. Mat. 17, 25–45 (1957/1958)22. Magenes, E., Stampacchia, G.: I problemi al contorno per le equazioni differenziali di tipo ellittico. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 12, 247–358 (1958)