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DOI: 10.14529/mmp 170112
THE CONTRIBUTION OF ANGELO EAYINI IN TWENTY YEARS OF JOINT RESEARCH (1996 - 2016)
S. Romanelli, Universita' degli Studi di Bari Aldo Moro, Italy
During his long activity in research, Angelo Favini has given several relevant contributions in many fields of mathematics. In our long collaboration, testified by more than thirty joint papers, I highly appreciated his deep competence, his brilliant intuitions, his extraordinary knowledge of the contemporary literature, in addition to a special acumen in finding possible critical points to be clarified, or better solved, during the preparation of the papers.
Here follows a short survey of the main problems studied in our joint papers over the last twenty years, often in collaboration with other outstanding mathematicians.
(1) At the beginning of 1990s there was a long standing open problem concerning the existence of analytic semigroups generated by second order elliptic differential operators degenerating at the boundary. It is well-known that the existence of a (Co) semigroup guarantees the wellposedness of the abstract Cauchy problem associated with the evolution equation governed by the generator, provided that the initial datum is in its domain. The analyticity of the semigroup provides, in addition, the best possible regularity of the solution, even by starting from the initial datum in the ambient space. As already known in Probability, or in Approximation Theory, in the space of continuous functions C(Q) (here Q is a bounded open subset of RN with sufficiently smooth boundary dQ), a possibly degenerate on the boundary, second order elliptic operator A can be naturally equipped with the so-called Wentzell boundary condition Au = 0, introduced in the one-dimensional case by W. Feller in his pioneer work [7] (see also [6]), and in the multidimensional case by A.D. Wentzell [38]. For Wentzell boundary conditions in different spaces see e.g. [33,37]. Observe that, in an evolution equation ut = Au, replacing Au by ut in Wentzell boundary condition reveals that, under suitable regularity assumptions on the elements of the domain
Au
with respect to t along the boundary. An easy example of degenerate elliptic second order differential operator on the space C[0,1] equipped with Wentzell boundary conditions is Aju = xj(1 — x)ju , j > 0, well-known also for governing some evolution problems in genetics. In [1] the existence of analytic semigroups generated by Aj was proved for j > 2 on C[0,1] by using different types of boundary conditions, including Wentzell's ones. After that, much attention was paid to this problem and relevant contributions by A. Favini et al. appeared in different papers, where, in addition to the space of continuous functions, as in [27,30,31], also Lp spaces, with or without weights, and Sobolev spaces were considered, see e.g. [2,20,28,29]. Results concerning the wave equation with Wentzell boundary conditions in C[0,1] were also obtained in [16].
(2) In the real-valued space C(Q) other interesting problems concern the existence of Feller semigroups generated by second-order elliptic differential operators degenerating on dQ and equipped with different boundary conditions, including Wentzell's ones. In this framework, some enlightening results were obtained by K. Taira, A. Favini and S. Romanelli in the papers [34-36], under suitable regularity assumptions on the coefficients of the operator and on dQ.
(3) If A denotes a second order, linear or nonlinear, elliptic differential operator, a general boundary condition including both Robin (and Dirichlet, Neumann) boundary conditions b+ cu = 0, relevant for Lp spaces, and Wentzell boundary conditions Au = 0, relevant for spaces of continuous functions, arises naturally as general Wentzell boundary condition given by
(GWBC) aAu(x) + b^M + m(x) = 0, x E dq,
on
where (a,b,c) = (0, 0, 0) and |n is the outer normal derivative of u. In [13] A. Favini, G.E. Goldstein, J.A. Goldstein and S. Romanelli introduced (GWBC) for the operator Au = au'' with domain
D(A) = {u E C[0,1] n C2(0,1) : Au E C[0,1], usatisfies (GWBC)}
in C[0,1]. Note that, in this case, (GWBC) consists of two conditions and reads as
a3 Au(j) + (—1)j b3 u'(j) + cj u(j) = 0, j = 0,1,
with aj,bj,Cj real numbers.
Therein, under the assumptions that a E C(0,1) a > 0 and a e L1(0,1) (a can vanish at {0,1}) and suitable assumptions on aj,bj,Cj, j = 0,1, the authors stated sufficient conditions yielding meaningful properties of the operator A in C[0,1], as density of the
A
and rn-dissipative whenever a0 = 1 = a1 and (—1)jbj < 0 and Cj > 0, with Cj = 0 if bj = 0, j = 0,1. More general nonlinear operators and nonlinear (GWBC) can also be considered (see e.g. [17]). Hence, the previous results provide extremely useful tools in the study of wellposedness for abstract Cauchy problems associated with a wide class of evolution problems.
(4) The wellposedness of abstract Cauchy problems with (GWBC) in Lp spaces, 1 < p < to requires the introduction of special spaces, as, for instance, discussed in the simple
Au = u L2(0, 1)
order to study (Co)-semigroups governed by operators equipped with (GWBC) in spaces of Lp-type, a new approach is necessary. In [18] A. Favini, J.A. Goldstein, G.R. Goldstein and
A
form Au = V • (aV), where a E C 1(Q) a is nonnegative and satisfies r := {x E dQ : a(x) > 0} = 0, and A is equipped with the following (GWBC)
du
Au + ¡3——+ yu = 0 on r, on
with 3,y nonnegative functions in Cl(3Q), ¡3 > 0. Note that, if we consider the evolution equation ut = Au equipped with (GWBC), under assumptions of sufficient regularity for u, then we can plug the term ut = Au in the boundary condition and obtain the condition ut + ¡3du + Yu = 0. Hence, the term Au corresponds to introduce a dynamic condition on the boundary. Coming back to the introduction of the new spaces, let us assume that r = dQ. For 1 < p < to, the correct Lp space to consider is Xp := Lp(Q, dy), where dy = dx|n © ^jf |dQ. Here dx denotes the Lebesgue measure on Q and af denotes the natural surface measure dS on dq with weight r p = to ^te space X^ can
ПЕРСОНАЛИИ
be identified with C(Q). One can interpret Xp, 1 < p < ж, as the соmpletion of C(Q) with respect to the norm || • ||xp given by \\U||xp = ^fQ \u\p dx + fdQ \u\p as^ p, where, if
u E C(Q), we consider U = (u\n,u\dn). In general, a member of Xp is a pair H = (f,g), where f E Lp(q,dx) and g E Lp(dq, ^^) and, for p < ж, f таУ n°t have a trace on OQ, and even if f does, this trace needs not equal g. For p = 2, X2 is a Hilbert space equipped with the inner product < H1,H2 >x2 = < fi,f2 >l2(q) + < gi,g2 >ь2(эп as), with Hi = (fi,gi) E X2, i = 1, 2. Under previous assumptions, in [18], the authors proved that there exists a (Co) semigroup generated by the closure of the realization of A in Xp, 1 < p < ж, and this semigroup is analytic if 1 < p < ж. In addition, A is essentially X2 A
A
(GWBC). See [11]. It is very important to point out that there is a precise physical derivation of (GWBC), as it was shown in details by G.E. Goldstein in [32]. For instance, in the case of the heat equation, in the evaluation of the total heat content of the region, the use of (GWBC) allows to consider also the contribution of a heat source located on the boundary, while in all usual approaches concerning the traditional boundary conditions, this type of contribution appears completely neglected. All the above results opened the way to a more and more wide literature and to a diffused interest of the international community of mathematicians, including those ones involved in the study of dynamical boundary conditions. Thus, in my opinion, the above results represent the milestone of our scientific collaboration.
(5) In the last years, starting from the paper [11], A. Favini, G.E. Goldstein, J.A. Goldstein, E. Obrecht and S. Romanelli found a more complete formulation of (GWBC) in this form
adu
(GGWBC) Au + ¡3— + yu - q/3albu = 0, on дQ,
on
including the term with the Laplace - Beltrami operator alb having nonnegative constant qA
A
A X2
(GGWBC)
analytic semigroup in X2 with angle of analyticity In the same papers [11,12], under C™ regularity assumptions on дQ, on all the coeflicients of A and on в,!-, we also showed that the closure of the realization of A in Xp, 1 < p < ж, is m-dissipative and generates an analytic semigroup having sector of analyticity depending on the moduli of ellipticity A
papers by A. Favini, G.E. Goldstein, J.A. Goldstein, S. Eomanelli and other coauthors, as G.M. Coclite, C.G. Gal and E. Obrecht, see e.g. [3-5,8,12,21,25], additional interesting results were obtained with respect to continuous dependence of the solutions from the boundary coefficients, hyperbolic problems, nonautonomous wave equation, nonlinear operators, nonsymmetric operators, unbounded domains.
(6) Extensions of some previous results described in (5) can be obtained also in the
X2
papers Wentzell boundary conditions together with lower order boundary conditions are
associated with this type of operators. For instance, in X2 = L (q,dx) © L (dQ, —-), a fourth order operator of the type Au = a(aa)u, equipped with both boundary conditions Au + P + Yu = 0 and Au = 0 is essentially self-adjoint and bounded below, provided that a E C4(Q), a > 0 dQ is C4 and E C3+e(dQ). A more complete classification of (GWBC) for the fourth order differentia 1 operator Au = u'm acting on X2, with Q = (0,1), can be found in [23] where we obtained a characterization of the symmetry and sufficient
A
References
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ПЕРСОНАЛИИ
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Received January 17, 2017
