GENERALIZED WENTZELL BOUNDARY CONDITIONS FOR SECOND ORDER OPERATORS WITH INTERIOR DEGENERACY

. We consider operators in divergence form, A 1 u = ( au (cid:48) ) (cid:48) and in nondivergence form, A 2 u = au (cid:48)(cid:48) , provided that the coeﬃcient a vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators A 1 and A 2 , equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of L 2 type.


1.
Introduction. It is well known that degenerate parabolic equations are widely used as mathematical models in the applied sciences to describe the evolution in time of a given system. For this reason, in recent years an increasing interest has been devoted to the study of second order differential degenerate operators in divergence or in nondivergence form. A wide non exhaustive description of them can be found in [9,Introduction] and in the references therein. In particular, operators of the type A 1 u := (au ) , A 2 u := au with suitable domains involving different boundary conditions, arise in a natural way in several contexts as aeronautics (Crocco equation), physics (boundary layer models), genetics (Wright-Fisher and Fleming-Viot models), mathematical finance (Black-Merton-Scholes models).
Here the novelty is that we deal with existence and regularity of solutions of Cauchy problems associated with parabolic equations having coefficients which degenerate in the interior of the spatial domain and satisfy general Wentzell boundary conditions in spaces of L 2 -type.
To our best knowledge, [16] is the first paper treating the existence of a solution for the Cauchy problem associated to a parabolic equation which degenerates in the interior of the spatial domain in the space L 2 (0, 1), while in [9] both the degenerate operators A 1 and A 2 in the space L 2 (0, 1), with or without weight, were 2. Basic assumptions and preliminary results. In the following we will introduce the notions of weak and strong degeneracy for a real-valued function a defined on the interval [0, 1].
Accordingly, we will define suitable weighted spaces and prove some formulas of Green type. These results will play a key role for the study of the operators A 1 and A 2 considered in Sections 3-5. Example 2.2. We can take a(x) = |x − x 0 | α , 0 < α < 1, as an example of a weakly degenerate function.
For any weakly degenerate a ∈ C[0, 1], let us introduce the following weighted spaces: The following Green formulae are analogous to those proved in [9]. Lemma 2.4. If a is weakly degenerate, then (i): for all (u, v) ∈ H 2 a (0, 1) × H 1 a (0, 1): The proof of Lemma 2.4 is given in the Appendix. Now let us introduce another notion of interior degeneracy.
(ii) The inclusion Z ⊆ H 2 a (0, 1) is obvious. Conversely, if u ∈ H 2 a (0, 1), then we only need to show that (au )(x 0 ) = 0. Similar arguments as in the proof of (i) imply that lim x→x − 0 (au )(x) = L ∈ R. If L = 0, then there exists C > 0 such that for all x in a left neighborhood of x 0 , x = x 0 . It follows that for all x in a left neighborhood of x 0 , x = x 0 . Hence |(a(u ) 2 )(x)| ≥ C 2 a(x) for all x in a left neighborhood of x 0 , x = x 0 . But a is strongly degenerate, so 1 a ∈ L 1 (0, 1) and thus √ au ∈ L 2 (0, 1). Hence L = 0. Analogously, one can prove that lim x→x + 0 (au )(x) = 0 and thus au can be extended by continuity at x 0 setting (au )(x 0 ) = 0.
We point out that Proposition 2.7 is based on the following Lemma 2.8. (see [9, Lemma 2.5]) If a is strongly degenerate, then for all u ∈ Z we have that for all x ∈ [0, 1].
As for the weakly degenerate case and using the previous characterization, we can prove the following Green's formulae. (See Appendix.) Lemma 2.9. If a is strongly degenerate, then Similar arguments as in [9, Propositions 3.6, 3.8] allow to characterize the spaces H 1 1 a (0, 1) and H 2 1 a (0, 1). Let us make the following additional assumption on a.
Hypothesis 2.1 There exists a positive constant K such that   3. Operators in divergence form with (GWBC): The weakly degenerate case. Let us fix β j , γ j ∈ R, j = 0, 1, such that β j > 0 and γ j ≥ 0, j = 0, 1. Consider a weakly or strongly degenerate function a, define the operator in divergence form and the space associated with (GWBC) (see [8]). Here dx denotes the Lebesgue measure on (0, 1), β = (β 0 , β 1 ), and adS β denotes the natural (Dirac) measure dS on {0, 1} with weight a β . More precisely, X µ is a Hilbert space with respect to the inner product given by (1))), and χ (0,1) is the characteristic function of the interval (0, 1). Hence X µ is equipped with the norm defined by for any f ∈ X µ , provided that f is written as (f χ (0,1) , (f (0), f (1))). We now define the weighted spaces and H 2 a (0, 1) := {u ∈ H 1 a (0, 1)| au ∈ H 1 (0, 1)}, endowed with the norms defined, respectively, by a (0, 1). Further, let us define the domain of A 1 to be the following subspace of X µ : 1}. Now we are ready for the main results of this Section. Theorem 3.2. If a is weakly degenerate, then the operator A 1 with domain D(A 1 ) is nonpositive and selfadjoint on X µ .
Moreover, W 1,a (0, 1) is dense in X µ by using the following argument: take u ∈ X µ and define u n : In order to show that A 1 is nonpositive and selfadjoint it suffices to prove that A 1 is symmetric, nonpositive and (I − A 1 )(D(A 1 )) = X µ (see e.g. [1, Theorem B.14] or [15]).
A 1 is symmetric.
By using Lemma 2.4(i) and (GWBC), for any u, v ∈ D(A 1 ), one has For any u ∈ D(A 1 ), according to the previous calculations, one has Observe that H 1 a (0, 1) is a Hilbert space with respect to the inner product for any u, v ∈ H 1 a (0, 1). Moreover, we have that From (3.2), the above equality means that Thus the weak derivative (au ) in this context exists and (au ) ∈ L 2 (0, 1). Hence u ∈ D(A 1 ) and Thanks to Theorem 3.2, one has that the problem is wellposed in the sense of Theorem 3.4 below. But first we recall the following definition.
Theorem 3.4. Assume that a is weakly degenerate. Then for all h ∈ L 2 (0, T ; X µ ) and u 0 ∈ X µ , there exists a unique weak solution u of (3.4) such that where K is a positive constant depending on T, a, β.
Proof. The assertion concerning the assumption u 0 ∈ X µ and the regularity of the solution u when u 0 ∈ D(A 1 ) is a consequence of the results in [2] and of [6, Lemma 4.1.5 and Proposition 4.3.9]. We only need to prove (3.5). Now let us fix u 0 ∈ D(A 1 ) and consider that the corresponding weak solution u is in C([0, T ]; D(A 1 )) ∩ C 1 ([0, T ]; X µ ). In the differential equation of (3.4) take the inner product in X µ of each term by u(t), for any t ∈ (0, T ). The result is and the regularity of u(t) implies that also u x (·, 0) and u x (·, 1) are in L 2 (0, T ). Hence we deduce that By Gronwall's Lemma, for any t ∈ [0, T ] one has Thus, there exists a positive constant C such that Observe that, by integrating the second inequality of (3.6) over (0, T ) and using (3.7), we have for a suitable positive constant C. We can conclude that there exists a positive constant K such that where K depends on T, a, β. Then the assertion follows.  In analogy with Proposition 2.7 one has the following   Proof. Let us introduce the space Then, analogous arguments as in Theorem 3.1 imply that W 1,as (0, 1) is dense in X µ and, hence, D(A 1 ) is dense in X µ . Moreover, as a consequence of Lemma 2.3 (i) and (GWBC), by arguing as in Theorem 3.1, one can show that A 1 is symmetric and nonpositive. In order to prove that I − A 1 is surjective, observe that W 1,as (0, 1) is a Hilbert space with respect to the inner product for any u, v ∈ W 1,as (0, 1). Notice that where (W 1,as (0, 1)) * is the dual space of W 1,as (0, 1)) with respect to X µ . Let f ∈ X µ and define F : W 1,as (0, 1) → R such that From W 1,as (0, 1) → X µ it follows that F ∈ (W 1,as (0, 1)) * . Hence, by the Riesz's Theorem, there exists a unique u ∈ W 1,as (0, 1) such that for any v ∈ W 1,as (0, 1) we have The above equality means that for all v ∈ W 1,as (0, 1). Let us denote by C ∞ c ((0, 1)\{x 0 }) the space of all C ∞ (0, 1) functions that vanish in a neighborhood of x 0 , with compact support in (0, 1) Thus the weak derivative (au ) in this context exists and (au ) ∈ L 2 (0, 1). Hence, u ∈ D(A 1 ) and u − A 1 u = f.  Thus Since u ∈ D, then (au ) u ∈ L 1 (0, 1). Hence there exists since no integrability is known about a(u ) 2 and such a limit could be −∞. If L = 0, there exists C > 0 such that for all x in a right neighborhood of x 0 , x = x 0 . Thus by (2.2) there exists C 1 > 0 such that for all x in a right neighborhood of x 0 , x = x 0 . This implies that u ∈ L 2 (0, 1) and thus u ∈ X µ . Hence L = 0 and Since (au ) u ∈ L 1 (0, 1), then √ au ∈ L 2 (0, 1). Thus √ au ∈ X µ and hence, D ⊆ D(A 1 ).
We point out the fact that the condition 1 a ∈ L 1 (0, 1) is crucial to prove the previous characterization.
Let us now introduce the following spaces Let us define the domain of A 2 as follows As a consequence of the results in Section 2, one has the next result.
Theorem 5.2. If a is weakly degenerate, then the operator (A 2 , D(A 2 )) is selfadjoint and nonpositive on Y µ .
A 2 is symmetric.
By Lemma 2.4(ii), for any u, v ∈ D(A 2 ), one has A 2 is nonpositive.
For any u ∈ D(A 2 ), according to the previous calculations, one has Similar arguments as in Theorem 3.2 can be applied.
Further, if a is strongly degenerate analogous results as in Proposition 2.10 hold, provided that one replaces the spaces H 1 It is evident that Similar arguments as in the proof of Theorem 5.1 show that A 2 is a symmetric and nonpositive operator. Let us show that for any u, v ∈ W 1, 1 a (0, 1). Moreover, it follows that a (0, 1)) * is the dual space with respect to Y µ . Let f ∈ Y µ and define F : W 1, 1 a (0, 1) → R as follows Since W 1, 1 a (0, 1) → Y µ , then F ∈ (W 1, 1 a (0, 1)) * . As a consequence, by Riesz's Theorem, there exists a unique u ∈ W 1, 1 a (0, 1) such that for any v ∈ W 1, 1 a (0, 1) From (5.1), the above equality is equivalent to write for all v ∈ W 1, 1 a (0, 1). As in Section 4, let us denote by C ∞ c ((0, 1) \ {x 0 }) the space of C ∞ (0, 1) functions that vanish in a neighborhood of x 0 , with compact support in (0, 1) Thus the weak derivative u in this context exists and au ∈ L 2 1 a (0, 1). Hence, u ∈ D(A 2 ), and, according to (5.3) and Lemma 2.9 (ii), we have that

FRAGNELLI, RUIZ GOLDSTEIN, GOLDSTEIN, MININNI AND ROMANELLI
As a consequence of Theorems 5.2 and 5.3, one has that A 2 is the infinitesimal generator of a strongly continuous semigroup on Y µ . Hence, the problem x ∈ (0, 1), is well-posed in the sense of evolution operator theory. In particular, the following theorem holds.
Theorem 5.4. Assume that a is weakly degenerate (resp. strongly degenerate and the Hypothesis 2.1 is satisfied). Then for all h ∈ L 2 (0, T ; Y µ ) and u 0 ∈ Y µ , there exists a unique weak solution where the constant C in (5.5) depends on T, a, β, but is independent of u o and h.
Appendix. In the following we will give the proofs of Lemmas 2.4 and 2.9.