1) The document discusses Enrico Magenes' early research in partial differential equations in the 1950s, applying Picone's method to transform boundary value problems into integral equations. 2) It describes Magenes' collaboration with G. Stampacchia at the University of Genoa in the late 1950s, where they studied works by Schwartz and others on weak solutions and Sobolev spaces and published an influential paper applying these concepts. 3) It outlines Magenes' long collaboration with J.-L. Lions in the 1960s, where they developed a general framework for defining weak solutions and traces for non-homogeneous boundary value problems using duality and distribution theory.

1. Some Aspects of the Research of Enrico
Magenes in Partial Differential Equations
Giuseppe Geymonat
Abstract The author traces the initial stage of Enrico Magenes’s research, with a
particular emphasis on his work in Partial Differential Equations. The very fruitful
collaborations with G. Stampacchia and J.-L. Lions are clearly presented.
1 The Beginnings in Modena
The first researches of Enrico Magenes in Partial Differential Equations date to
1952, [14, 15] (and in the same year he became professor of Mathematical Analysis
at the University of Modena). Their argument is the application to the heat equation
of a method that in the Italian School is called “Picone’s Method”. The basic idea
of the method is to transform the boundary value problem into a system of integral
equations of Fischer-Riesz type. This idea was introduced by Picone around 1935
and 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-homogeneous
Dirichlet 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 and
γ0 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 the
so-called co-normal derivative (when A = , then A = and ν = n, the outgoing
normal to Γ ). The Green formula states
∂u ∂w
A(u)wdx − uA (w)dx = w−u dΓ. (2)
Ω Ω Γ ∂ν ∂ν
Hence, 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 vector
G. Geymonat (B)
LMS, Laboratoire de Mécanique des Solides UMR 7649, École Polytechnique, Route de Saclay,
91128 Palaiseau Cedex, France
e-mail: giuseppe.geymonat@lms.polytechnique.fr
F. Brezzi et al. (eds.), Analysis and Numerics of Partial Differential Equations, 5
Springer INdAM Series 4, DOI 10.1007/978-88-470-2592-9_2,
© Springer-Verlag Italia 2013

2. 6 G. Geymonat
(u, ∂u ) with u the solution of (1) is reduced to the solution of a system of linear
∂ν
equations of Fischer-Riesz type.
The difficulty was naturally to find such a sequence, to prove its completeness (in
a suitable functional space) and hence the space where the corresponding system is
solvable 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 the
fundamental solution of A (w) = 0 (moreover, such a fundamental function can be
chosen 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 can
take 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 of
the simple and double layer potentials, appearing in (3) and (4). See for instance
Fichera’s paper [3], where many properties are studied, and in particular results of
completeness are proved. (The modern potential theory studies the fine properties
of 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 ),
∂t
whose 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 Ω
Following 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

3. Research in PDE 7
equation E(u) = f , assuming the boundary value in a suitable way and represented
by 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 the
data, 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 using
potential theory not only of Magenes (see e.g. [18, 19]) but also of Fichera, Miranda,
Stampacchia, . . .
2 The Years in Genoa with G. Stampacchia
From the historical point of view, these researches show the change of perspective
that 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 gives
a typical account (see [20]). Breve esposizione e raffronto dei più recenti sviluppi
della teoria dei problemi al contorno misti per le equazioni alle derivate parziali
lineari 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 order
elliptic 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 of
Genoa, where he had G. Stampacchia as colleague. Stampacchia was a very good
friend of Magenes from their years as students at Scuola Normale, since both where
antifascist. Moreover, Magenes and Stampacchia were well aware of the fundamen-
tal change induced by the distribution theory and the Sobolev spaces in the calculus
of variations and in the study of partial differential equations, particularly in the
study 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 results
on 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 Spring
1958, J.-L. Lions gave at Genoa a series of talks on the mixed problems [5, 6], and
in June 1958 Magenes and Stampacchia completed a long paper [22], that would
have a fundamental influence on the Italian researches on elliptic partial differential
equations. Indeed, that paper gives a general presentation of the results obtained up
to that moment in France, United States, Sweden, Soviet Union by N. Aronszajn,

4. 8 G. Geymonat
F.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 Boundary
value 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 this
methodology in the Italian mathematical community, giving lectures in various uni-
versities (see e.g. [21]), for instance organizing with Stampacchia a CIME course
on 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 and
they started a long series of joint works [7–10], whose results were summarized
and fully developed in a series of books [11–13] translated in Russian, English and
Chinese.
3 The Collaboration with J.-L. Lions in the Study of Boundary
Value Problems
Following an idea of J. Hadamard, Courant and Hilbert (Methods of Mathematical
Physics, 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 spaces
where the problems are well-posed. The distribution theory and the Sobolev spaces
give 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 posed
in Sobolev spaces W m,2 (Ω) with m big enough. For non-homogeneous boundary
data, the situation was more difficult, since at first it was necessary to give a good
definition of the trace γ0 u of an element u ∈ W m,p (Ω) on Γ := ∂Ω. The good
definitions 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
1
its boundary Γ ) by E. Gagliardo, J.-L. Lions, and P.I. Lizorkin, G. Prodi, . . . . In
particular it was proved that the trace operator cannot be continuously defined on
L2 (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 sometimes
of negative order. Therefore, it is necessary to define a weak or generalized solution
of a non-homogeneous boundary value problem and hence, to give a good definition
of trace in a weak sense.

5. Research in PDE 9
Inspired by the theory of distributions, Lions and Magenes [7–10] tackled the
problem 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 domain
Ω ⊂ RN with smooth boundary Γ , and Bγ0 is a linear differential operator with
smooth 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]), the
map A : E(Ω) → F (Ω) × G(Γ ) is an isomorphism (for simplicity, and in general a
finite index operator) between the Sobolev spaces E(Ω) and F (Ω) × G(Γ ), where
these 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 to
the 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), where
X(Ω) is a subspace of E(Ω).
By transposition, for every linear and continuous form L(v) on X(Ω), there
exists 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,
m
Lions and Magenes chose L = L1 + L2 in such a way that L1 gives rise to the
equation 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 the
most natural way.
Perhaps the most interesting contribution of Lions and Magenes was the optimal
choice of L2 . It was obtained thanks to a clever use of the Green formula, that allows
to naturally define the traces of every element u ∈ (F0 (Ω)) , such that A u belongs
to 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 ).

6. 10 G. Geymonat
4. 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 spaces
and 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)

7. Research in PDE 11
16. 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)
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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)
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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)