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1 MAXIMAL REGULARITY IN L ) FOR A CLASS OF ELLIPTIC OPERATORS WITH UNBOUNDED COEFFICIENTS Giovanni Cuini Diartimento di Matematica U. Dini Università degli Studi di Firenze, Viale Morgagni 67/A, I-5034 Firenze Italy) cuini@math.unifi.it Simona Fornaro Diartimento di Matematica E. De Giorgi Università degli Studi di Lecce, Via er Arnesano, C.P. 93, I-7300 Lecce Italy) sfornaro@ilenic.unile.it Abstract. Strongly ellitic differential oerators with ossibly) unbounded lower order coefficients are shown to generate C 0-semigrous on L ), < < +. An exlicit characterization of the domain is given.. Introduction Linear ellitic oerators with regular and bounded coefficients have nowadays a satisfactory theory including existence, uniqueness and regularity for the solutions to the corresonding equations in several Banach saces, such as L saces, Hölder saces and so on. On the other hand, ellitic oerators with unbounded coefficients are still object of intensive investigation, as the recent literature shows. The increasing interest towards such class of oerators is due also to the analytic treatment of stochastic differential equations. In this aer we consider the following class of second order ellitic oerators.) Au := A 0 u F, Du V u, where A 0 u := D i q ij D j u). As usual, we will refer to F and V as the drift and the otential term, resectively, and neither F nor V will be assumed to be bounded. Our aim is to rove a generation result for A in L ) < < + ) with resect to the Lebesgue measure, roviding an exlicit descrition of the domain of the generator. Precisely, we show that such domain is the intersection of the domains of each added of A as in.). There are several aroaches to show that ellitic oerators with unbounded coefficients generate strongly continuous semigrous in L. On this subject we mention [2], [3], [6], [], [3], [4] and the list of references therein. However only some of them give a recise descrition of the domain. In [4], [5] and in [5] only the secial case of = 2 is considered. This aer gets insiration essentially 2000 Mathematics Subject Classification. 35J70, 35K5, 35K65. Acknowledgements. The authors wish to thank Giorgio Metafune, Diego Pallara and Vincenzo Vesri for many helful conversations and suggestions.

2 2 G. Cuini - S. Fornaro from [3] and [4]. In [3] the case V = 0 and F globally Lischitz continuous is considered. It is roved that under the assumtion { F, Dq ij L ), i, j =,..., } N, the corresonding oerator A, endowed with the domain u W 2, ) : F, Du L ), generates a C 0 -semigrou in L ), < < +. Here, the characterization of the domain follows from regularity results for the solution to the non homogenous Cauchy roblem associated with A. In [4] a second order oerator in the general form.) is considered and the descrition of the domain of the generator is given in L ) assuming the conditions DV γv 3/2, F κv /2 and divf βv. We observe that the first two assumtions are the same of Cannarsa and Vesri in [3], whereas the last one relaces an additional bound on the constant κ assumed in [3]. In [4], with a more direct aroach, { it is roved that A generates } an analytic semigrou on L ), < < +, with domain u W 2, ) : V u L ). The cases = and = + are also considered. In this aer we rove that if D, D ), with < < +, is the Banach sace defined as D := {u W 2, ) : F, Du L ), V u L )}, u D := u W 2, ) + F, Du L ) + V u L ), then A, D ) generates a C 0 -semigrou in L ), if suitable growth conditions on F, V and their first order derivatives are assumed. As a by-roduct, one can deduce regularity results for the solutions of the ellitic equation associated with A. The aer is structured as follows. In Section 2 we establish the notation used throughout the aer and we state our assumtions and our main results. We searately consider the articular case = 2 and the general case < < +, owing to the two different aroaches used, which require different sets of hyotheses. Of course, those for = 2 are weaker then those for an arbitrary. We remark that for suitable choices of the arameters involved, our framework covers [3] or [4]. Thus, our results can be seen as a continuous interolation between them. We also cover new cases. We refer to Section 2 for further details and comments on the assumtions. In Section 3 we consider, as in [3], the case where F is globally Lischitz, but here we focus on an a riori estimate for the second order derivatives of a test function, recising the deendence of the constants obtained. This fact will be crucial for the sequel. In order to show that A, D ) generates a semigrou, in Section 4 we rove a riori estimates for Du and V u with resect to the L norm, for every, ) and every test function u. To do this we use basically integrations by arts and other elementary tools. In the articular case = 2, we also get an estimate for the second order derivatives of u. In Section 5 we deal with the generation result for = 2 see Theorem 2.). As a simle consequence of the estimates reviously roved, A, D 2 ) is closed and quasi dissiative. To rove that the oerator λ A : D 2 L 2 ) is surjective for λ large enough we find the solution of the equation λu Au = f in the whole sace as the limit of a sequence of solutions of the same equation in balls with increasing radii and Dirichlet boundary conditions. The generation result for the case < < + see Theorem 2.2) is roved in Section 6. In this framework the assumtions are more restrictive than before, since the variational method of Section 4 fails to estimate the second order derivatives of u Cc ). Therefore, we use a different technique which works under stronger hyotheses. The idea is the following. We use a change of variable, determined by V, together with a loocalization argument in order to obtain a family of oerators {A xn } n N with globally Lischitz drift coefficients and bounded otentials. To each oerator A xn we aly the estimate of Section 3, thus obtaining local estimates in the original setting, with uniform constants. Using a covering argument we get the global estimate we were looking for. Once that the estimate on the second order derivatives is obtained, the surjectivity of

3 Maximal regularity in L ) for a class of ellitic oerators... 3 the oerator λ A follows by aroximating A with a family of oerators satisfying the assumtions of Section 3. Finally, in Section 7 we describe some roerties of the above semigrous. We rove that they are ositive, not analytic in general, consistent with resect to. Moreover if V tends to + as x +, then A, D ) has comact resolvent. 2. Notation and statement of the main results Throughout this aer Cc ) is the sace of real-valued C -functions on with comact suort and Cb RN ) is the sace of real-valued functions on, which are bounded and continuous together with their first order derivatives. We denote by the su-norm in and by st φ the suort of a given function φ. For and k N, L ) and W k, ) are the usual Lebesgue and Sobolev saces, resectively. The norm of L ) is denoted by and k, denotes that of W k, ). Given a function u on, we denote its gradient and its Hessian matrix by Du and D 2 u, resectively. Moreover, we set Du 2 = D i u) 2, D 2 u 2 = D ij u) 2, i= where, clearly, D i = D xi and D ij = D xi x j. The ball in centered in x with radius r > 0 is indicated by Bx, r). To shorten the notation, if x = 0 we will write B r instead of B0, r). If J is a set, card J is its cardinality and χ J is the characteristic function of J, that is χ J x) = if x J and χ J x) = 0 if x J. In the following qx) = q ij x)) is a N N symmetric real matrix such that q ij C b RN ) and 2.) qx)ξ, ξ := q ij x) ξ i ξ j ν ξ 2, ν > 0, for every x, ξ. Moreover, we consider F C ; ) and V C ) and we assume that V is bounded from below. Without loss of generality, we suose that V. We deal with the ellitic oerator 2.2) Au := A 0 u F, Du V u, where A 0 ux) := N D i q ij x)d j ux)). For < < +, we define the sace D, D ) as 2.3) 2.4) D := {u W 2, ) : F, Du L ), V u L )}, u D := u 2, + F, Du + V u. We endow D also with the grah norm of the oerator A, namely u A := Au + u. In the case = 2, besides the revious assumtions on the coefficients, we require that the following growth conditions hold H) DV αv 3/2 + c α, H2) divf βv + c β, D i F j x)ξ i ξ j τ V x) ξ 2 c τ ξ 2, ξ, x, H3) F, DV γv 2 + c γ,

4 4 G. Cuini - S. Fornaro H4) F x) θ + x 2 ) /2 V x), with α, β, γ, τ, θ > 0 and c α, c β, c γ, c τ 0 satisfying 2.5) β 2 τ > 0, and M 2.6) 4 α2 + β 2 + γ 2 <, where M := su x max ξ = qx)ξ, ξ. We note that the second inequality in H2) is a dissiativity condition for the function F. The following generation result holds. Theorem 2. =2). Suose that H), H2), H3), H4), 2.5) and 2.6) hold. Then the oerator A, D 2 ) generates a C 0 -semigrou on L 2 ). If c β = 0, then the semigrou is contractive. In Section 6 we rove an analogous result in the general case >. To this aim we use a different technique, which works under more restrictive assumtions on the coefficients of A. Precisely, we relace assumtions H), H2) and H4) with the following ones H ) DV x) α V 2 σ x) + x 2 ) µ/2, H2 ) DF N βv + c β ), H4 ) F x) θ + x 2 ) µ/2 V σ x), resectively, where DF denotes the Jacobian matrix of F and DF 2 = N k,i= D k F i 2, α, β, θ > 0, c β 0, 2 σ and 0 µ. Moreover, we suose that for every x RN H5) F x), Dq ij x) κ V x) + c κ, holds, with constants κ > 0 and c κ 0. Analogously to the case = 2, also in this case a smallness condition on the coefficients is required. Let Then we assume that 2.7) ω := The following generation result holds. M 4 )α2, ) if σ, µ) = 2, 0, 0, otherwise. ω + 2 β + Nαθ + αθ <, if < < 2, ω + 2 β + ) Nαθ + ) <, if 2. N Theorem 2.2 <<+ ). Suose that H ), H2 ), H4 ), H5) and 2.7) are satisfied, for some < <. Then the oerator A, D ) generates a C 0 -semigrou on L ), which turns out to be contractive if c β = 0. Remark 2.3. We observe that in 2.7) the condition for 2 imlies the condition for < < 2, since β + Nαθ 2 + αθ 2 β + ) Nαθ + ), >. N

5 Maximal regularity in L ) for a class of ellitic oerators... 5 Moreover, we note that when = 2, 2.7) is not equivalent to 2.6), but it is stronger. This fact relies on the different technique emloyed in the general case and, in articular, on the fact that we need that other suitable oerators verify our assumtions. For further details we refer to Section 6. In any case, the two methods yields the same semigrou in L 2 ). Finally, we oint out that in Theorem 2.2 we do not exlicitly assume H3), since H ) and H4 ) imly 2.8) F, DV αθv 2. Remark 2.4. Hyothesis H) is essential to determine the domain. In fact in [4, Examle 3.7] the authors resent a Schrödinger oerator A = V on L 2 R 3 ) such that H) holds with a too large constant α and the domain is not W 2,2 R 3 ) DV ). Moreover in [4] it is observed that H) holds for examle for any olynomial whose homogenous art of maximal degree is ositive definite. However H) fails for the function + x 2 y 2. Remark 2.5. We note that making articular choices of the arameters µ and σ, we may cover cases already known or discuss new ones. For examle, if µ = 0 and σ = 2, then we get exactly the framework of [4] F θv /2, DV αv 3/2 and therefore of [3]. If we take V constant, then we reduce to the case where F is globally Lischitz studied in [3]. Setting µ = 0 and σ = we have the case F θv, DV αv, which, according to our knowledge, seems to be new. From the second condition above, one deduces that V grows at most exonentially. Observe, however, that the exonent α is small, by 2.7). In any case, we can treat in this way olynomials V as in Remark 2.4. If we otimize assumtion H4 ) choosing µ = σ =, analogously to H4) in the case = 2, then H ) becomes DV x) α V x), which is much more restrictive than H). This shows that + x 2 ) /2 the cases = 2 and 2 are quite different. Such a difference is also confirmed by the fact that when = 2 we do not require any condition on Dq ij, F. The assumtions for 2 are determined but our aroach to estimate the second order derivatives of a test function u in terms of u and Au. The idea is to get first local estimates. To this aim we change variables and localize the equation Au = f in certain balls Bx 0, rx 0 )). The new oerator roduced by this technique see 6.4)) has a globally Lischitz drift term and a bounded otential. The radius rx 0 ) has to grow at most linearly with resect to x 0 in order to use a covering argument and to obtain global estimates. So, roughly seaking, we must require that rx 0 ) + x 0 and that V x) is close to V x 0 ) if x x 0 < rx 0 ). This is exactly guaranteed by assumtions H4 ) see 6.2)) and H ) see Lemma 6.3). The Lischitz continuity of the transformed drift coefficient follows from H2 ). All the details are given in Section Oerators with globally Lischitz drift coefficients In this section we collect all the results concerning oerators with globally Lischitz drift coefficient and bounded otential term that will be used in the sequel. We use the same technique as in [3], but here we recise how the constants involved deend on the oerator. Let 3.) B = D i a ij D j ) b i D i c and assume that i=

6 6 G. Cuini - S. Fornaro i) a ij = a ji Cb RN ), N a ij ξ i ξ j ν ξ 2, ii) b = b,..., b N ) is globally Lischitz in, iii) c L ), iv) su x Da ij x), bx) < +, i, j =,..., N. The following a riori estimate is a crucial oint for our aims. Theorem 3.. There exists a constant C deending on, N, ν, a ij, Da ij, Da ij, b, c and the Lischitz constant of b, [b], such that for all u Cc ) 3.2) D 2 u dx C Bu + u ) dx. Proof. We slit the roof in two stes. Ste. We assume that the oerator B is written in the non-divergence form B = a ij D ij b i D i c i= and that b C 2 ; ) with bounded first and second order derivatives, besides assumtions i), ii), iii) and iv). Let u Cc ). Then u solves the equation D t u Bu = f in +, with f = Bu. Let us consider the ordinary Cauchy roblem in dξ 3.3) dt = bξ), t R ξ0) = x. Since b is globally Lischitz, for all x there is a unique global solution ξt, x) of 3.3) and the identity 3.4) x = ξt, ξ t, x)), t R, x holds. Moreover, from [2, Section 2.] it follows that if ξ x denotes the Jacobian matrix of the derivatives of ξ with resect to x, then 3.5) ξ x t, x) e t Db, t R, x ξ tx t, x) Db e t Db, t R, x t ξ xt, ξ t, x)) Db e 3 t Db, t R, x. With analogous notation we have also that 3.6) ξ xx t, x) e t Db e t Db ) D2 b, t R, x Db ξ x t, ξ t, x)) x e3 t Db e t Db ) D2 b, t R, x, i =,...N. i Db In articular, all the above functions are bounded in [ T, T ], for every T > 0. Finally, the matrix ξ x is invertible with determinant bounded away from zero in every stri [ T, T ]. Setting vt, y) = uξ t, y)), a straightforward comutation shows that D t v Bv = f, in +

7 with ft, y) = fξ t, y)) and Maximal regularity in L ) for a class of ellitic oerators... 7 B = ã ij t, y) = b i t, y) = h,k= h,k= ct, y) = cξ t, y)). ã ij t, y)d yi y j + b i t, y)d yi c, i= D xh ξ i t, ξ t, y))a hk ξ t, y))d xk ξ j t, ξ t, y)) D xh x k ξ i t, ξ t, y))a hk ξ t, y)), Since the coefficients a ij belong to C b RN ) and satisfy iv), then t, y) a ij ξ t, y)) is bounded and differentiable with bounded derivatives in [ T, T ]. Taking into account 3.5) and 3.6) it follows that for all t, y) [ T, T ] we have ã ij t, y) + D t ã ij t, y) + D yk ã ij t, y) + b i t, y) L, i, j, k =,...N, where L deends on T, N, a ij, Da ij, Da ij, b, Db, D 2 b. Moreover ã ij t, y)η i η j ν η 2, η, y, t [ T, T ], with ν deending on ν, T, Db. Finally, the modulus of continuity of ã ij deends only on T, N, a ij, Da ij, Da ij, b, Db, D 2 b. Therefore D t B is a uniformly arabolic oerator in +. Alying the classical L -estimates available from the theory of uniformly arabolic oerators see e.g. [9, Section IV.0]) we have that 3.7) /2 /2 D y vt, y) + D 2 yvt, y) )dy dt K ft, y) + vt, y) )dy dt where K deends on, N, ν, ã ij, Dã ij, D t ã ij, b i, c, hence on, N, ν, a ij, Da ij, Da ij, b, Db, D 2 b, c. In order to come back to the function u, we observe that, setting St)φ)x) = φξt, x)) then, for every fixed t, St) mas W 2, ) into itself and St)φ)x) dx α t) φy) dy, R N D x St)φ)x) dx α 2 t) D y φy) dy, R N DxSt)φ)x) 2 dx α 3 t) Dyφy) 2 + D y φy) )dy, with α t), α 2 t), α 3 t) deending on t,, N, su ξ x t, ) and α 3 t) deending also on su ξ xx t, ). It follows that t α i t), i =, 2, 3, are uniformly bounded in t in the interval [, ]. In the sequel we denote by α i the resective uer bound. Moreover, by 3.4) each St) is invertible with St) = S t). Now, recalling that u = St)v, for every t, we have Dxux) 2 dx α 3 Dyvt, 2 y) + D y vt, y) )dy.

8 8 G. Cuini - S. Fornaro Integrating from /2 to /2 and taking into account 3.7) we obtain Dxux) 2 dx α 3 K ft, y) + vt, y) )dy dt R N 2 α α 3 K fx) + ux) )dx, which is the claim. Ste 2. Take B in the general form 3.) and assume that the coefficients satisfy i), ii), iii) and iv). Then we can write N B = a ij D ij + D i a ij b j )D j c. j= i= Let η C c ), st η B, η 0, η = and set ˆb = b η. If we define ˆB = a ij D ij ˆbj D j c, then ˆB satisfies all the assumtions of the revious ste. Indeed, since b is Lischitz continuous, b ˆb is bounded: bx) ˆbx) bx) bx y) ηy)dy [b] y ηy)dy = c η [b]. Then and Da ij x), ˆbx) Da ij x), bx) + Da ij x), bx) ˆbx) Da ij, b + Da ij c η [b], Dˆb [b] j= D 2ˆb [b] Dη. From the first ste it follows that there exists a constant C deending on N,, ν, a ij, Da ij, a ij, b, [b], c such that for all u C c ) Therefore D 2 u C ˆBu + u ). D 2 u C Bu + Bu ˆBu + u ) C Bu + Du + u ), with C deending on the stated quantities. Using the interolatory estimate Du C 2 u /2 D 2 u /2 we conclude the roof. In [3] it is roved that the oerator B endowed with the domain D = {u W 2, ) : b, Du L )} generates a C 0 -semigrou on L ), < < +. Actually, in [3] c is equal to 0, but the same result easily extends to this case, since c is bounded. Arguing as in [3, Theorem 2.2], one can rove the following result. { } Proosition 3.2. If λ > λ, λ := su x div bx) cx), then, given f L ), there exists a unique solution u D of λu Bu = f and satisfies u λ λ ) f.

9 Maximal regularity in L ) for a class of ellitic oerators Estimates of V u and Du From now on, for clarity of exosition, we assume that c α = c β = c γ = c τ = 0 in conditions H), H2), H3). This is always ossible, keeing the same constants α, β, γ, τ, just relacing V with V + λ and choosing λ large enough this imlies ossibly different constants in the statements). In this section we rovide, as a reliminary ste, some a riori estimates for the solutions of the ellitic equation λu Au = f. Precisely, via integrations by arts and other elementary tools, we rove that for all u D, the L -norms of V u and Du may be estimated by the L -norms of Au and u itself, with constants indeendent of u. If = 2, we also deduce an analogous estimate for the second order derivatives of u. Let us first show that Cc ) is dense in D, D ), < < +, so that all our estimates will be roved on test-functions. Lemma 4.. Suose that H4) holds. Then C c ) is dense in D, D ). Proof. Let η be a cut-off function such that 0 η, η in B, st η B 2 and Dη 2 + D 2 η L. We write η n x) in lace of ηx/n). Suose that u D. It is easy to see that η n u u D, as n. In fact, η n u u in W 2, ) and V η n u V u in L ), by dominated convergence. Moreover, F, Dη n u) = η n F, Du + u F, Dη n. As before, the first term in the right hand side converges to F, Du in L ), as n goes to infinity. The second term tends to 0 since from H4) it follows that 4.) ) + 4n u F, Dη n dx L /2 θ V u 2 /2 B 2n \B n n 2 dx 5 /2 L /2 θ V u dx. \B n This shows that the set of functions in D having comact suort, denoted by D,c, is dense in D. Suose now that u D,c. A standard convolution argument shows the existence of a sequence of smooth functions with comact suort converging to u in D. Thus, the density of Cc ) in D, D ) follows. We state that, under rather weak assumtions, the oerator A, Cc )) is dissiative in L ), for any < < +. Lemma 4.2. Suose that 4.2) div F V. Then A, C c )) is dissiative in L ). Proof. We have to rove that for all λ > 0 and for all u C c 4.3) u λ λu Au. Let λ > 0 be fixed. If u C c ) one has ) we set u = u u 2 and recall that 4.4) Du ) = ) u 2 Du, D u ) = u Du. Set λu Au = f. Multilying both sides of this equation by u and integrating by arts, we obtain λ u + ) qdu, Du u 2 dx div F u dx + V u dx = fu dx.

10 0 G. Cuini - S. Fornaro By 2.) we get ) qdu, Du u 2 dx )ν Du 2 u 2 dx 0 and taking into account 4.2) it turns out that ) ) λ u fu dx f dx u dx. Multilying by u we get 4.3). Remark 4.3. It is noteworthy observing that if 4.2) holds, < 2 and u Cc ) then 4.5) Du c Au + u ) dx, where c = cν, ) > 0. In fact, from the roof of Lemma 4.2, with λ =, we deduce that 4.6) Du 2 u 2 dx ) ) u Au dx u dx ν ) R N c Au + u ) dx, where c = cν, ) > 0. If = 2, we are done. If < < 2, Young s inequality with exonent 2/ yields {u 0} Du dx = and 4.5) follows by 4.6). {u 0} Du u 2) 2 ) u 2) 2 dx c {u 0} Du 2 u 2 + u ) dx Remark 4.4. We note that condition H2 ), with c β = 0, together with 2.7) imlies condition 4.2), so that Lemma 4.2 still holds. If c β 0, then the same comutations of Lemma 4.2 show that A c β, C c )) is dissiative in L ), which means that oerator A, Cc )) is quasidissiative. Exlicitly, one has 4.7) u λ c β ) λ A)u, u C c ). In the following lemma we rove an estimate of the L -norm of V u. Lemma 4.5. Let < < +. Assume that H), H3) and 4.8) divf βv hold with 4.9) M 4 )α2 + β + γ <, where M := su x max ξ = qx)ξ, ξ. If u Cc ), then 4.0) V u dx c Au + u ) dx for some c > 0 deending only on, M, ν and on the constants in H), H3) and 4.8).

11 Maximal regularity in L ) for a class of ellitic oerators... Proof. Let u Cc ). We recall that if u = u u 2, then 4.4) holds. Integrating by arts one deduces A 0 u)v u dx = qdu, DV u ) dx R N = ) qdu, Du V u 2 dx ) qdu, DV V 2 u 2 u dx and V F, Du u dx = V F, D u ) dx = V divf u dx V 2 F, DV u dx. Thus, multilying 2.2) by V u and integrating, we obtain 4.) ) qdu, Du V u 2 dx + V u dx R N = Au)V u dx + V divf u dx + V 2 F, DV u dx ) qdu, DV V 2 u 2 u dx. Now, assumtions 4.8) and H3) imly 4.2) V divf u dx β V u dx and 4.3) V 2 F, DV u dx γ V u dx, resectively. By 2.) and H) the last term in 4.) can be estimated as follows 4.4) qdu, DV V 2 u 2 u dx qdu, Du /2 qdv, DV /2 V 2 u dx α M qdu, Du /2 V /2 u dx. Setting Q 2 := qdu, Du V u 2 dx and R 2 := V u dx, from Hölder s inequality it follows 4.5) qdu, Du /2 V /2 u dx QR. Thus, collecting 4.) 4.4) we obtain )Q 2 + β ) γ ) R 2 α ) MQR + Au)V u dx )Q 2 + )α2 M R Au)V u dx. Since Au)V u dx Au V u R R dx εr 2 + c ε Au dx, N N the thesis follows from 4.9) and by choosing ε small enough.

12 2 G. Cuini - S. Fornaro The next result rovides an L -estimate of V Du, with 2. In articular, since V, it extends estimate 4.5) to the case > 2. We exlicitly notice that we need a further assumtion on F, namely the dissiativity condition. Lemma 4.6. Let 2. Assume that H), H2), H3) and 4.9) hold and that β satisfies also the inequality 4.6) β τ > 0. If u Cc ), then 4.7) V Du dx + Du 2 D 2 u 2 dx c Au + u ) dx, with c deending on N,, ν, α, β, τ, M, Dq ij. Proof. We divide the roof in two stes: in the first ste we consider the sulementary assumtion that q ij C 2 ), in the second one we remove this condition via an aroximation rocedure. Ste. Suose that q ij C 2 ) Cb RN ), for every i, j N. Let u Cc ) and define f = λu Au, with λ > 0 to be chosen later. With a fixed k {,..., N}, we differentiate with resect to x k, so that 4.8) λd k u D i D k q ij D j u) D i q ij D jk u) + D k F i D i u i= + F i D ik u + ud k V + V D k u = D k f. i= Multilying 4.8) by D k u Du 2, summing over k =,..., N and integrating on we get 4.9) λ Du dx + I + I 2 + I 3 + I 4 + I 5 + V Du dx = Df, Du Du 2 dx, where I = I 2 = I 3 = I 4 = i,j,k= i,j,k= i,k= i,k= D i D k q ij D j u) D k u Du 2 dx, D i q ij D jk u)d k u Du 2 dx, D k F i D i u D k u Du 2 dx, F i D ik u D k u Du 2 dx, I 5 = DV, Du u Du 2 dx. Let us estimate the integrals above. Since t t t 2 is in C ; ), integrating by arts and alying Hölder s and Young s inequalities we have

13 Maximal regularity in L ) for a class of ellitic oerators... 3 I = D k q ij D j ud ik u Du 2 + 2) D k q ij D j ud k ud h ud ih u Du 4 i,j,k= i,j,k,h= c Du R D 2 u dx = c Du /2 Du 2)/2 D 2 u ) dx N c Du dx + c ε Du 2 D 2 u 2 dx, ε where c = c, N, Dq ij ) and ε > 0 is arbitrary. Consequently 4.20) I c Du dx c ε Du 2 D 2 u 2 dx. ε Assumtion 2.) allows to estimate the second integral, after an integration by arts; indeed I 2 = i,j,k= ν q ij D jk u D ik u Du 2 dx D 2 u 2 Du 2 dx + ν 2 4 D Du 2) 2 Du 4 dx. Since the last term is nonnegative we deduce that 4.2) I 2 ν Du 2 D 2 u 2 dx. From H2) it follows immediately that 4.22) I 3 τ V Du dx. As far as I 4 is concerned, integrating by arts, it turns out that I 4 = i,k= 2) which imlies by H2) that D i F i D k u) 2 Du 2 dx i,k,h= q ij D j Du 2 ) D i Du 2 ) Du 4 dx i,k= F i D k u) 2 D h u D ih u Du 4 dx = divf Du dx I 4 2)I ) I 4 = divf Du dx β F i D k u D ik u Du 2 dx V Du dx. Alying H) and Young s inequality, we get I 5 α V 3 2 u Du dx = α V u Du 2 2 )V 2 Du 2 ) dx α V u 2 Du 2 dx + εα V Du dx ε R N c 2 V u R dx + c 2 Du dx + εα V Du dx N

14 4 G. Cuini - S. Fornaro with c 2 = c 2 ε,, α). Then 4.24) I 5 c 2 V u R dx c 2 Du dx εα V Du dx. N We are left to estimate the integral in the right hand side in 4.9). arguing as before we obtain Integrating by arts and Df, Du Du 2 dx ) f Du 2 D hk u dx h,k= = ) f Du Du 2 D hk u dx h,k= c 3 f R 2 Du 2 dx + ε ) Du 2 D 2 u 2 dx, N with c 3 = c 3, N, ε). Alying Young s inequality we have finally Df, Du Du ) dx c 4 f R R dx + c 4 Du dx N N R N +ε ) Du 2 D 2 u 2 dx, with c 4 = c 4, N, ε). Collecting 4.20) 4.25) from 4.9) we obtain λ c ) ε c 2 c 4 Du dx R ) N + ν c + )ε Du 2 D 2 u 2 dx + β ) τ εα V Du dx R N c 2 V u R dx + c 4 f dx. N From 4.6) and 4.0), choosing first a small ε and then a large λ, we deduce that Du + V Du ) dx + Du 2 D 2 u 2 dx c Au + u ) dx, where the constant c deends on, N, ν, M, Dq ij and the constants in H), H2), H3). Ste 2. Let ϕ be a standard mollifier and set, as usual, ϕ ε x) = ε N ϕ x ε ). If q ε ij = q ij ϕ ε and A ε u = D i q ε ijd j u) F, Du V u, then by Ste, noticing that qij ε q ij, Dqij ε Dq ij and that qij ε ) satisfy 2.) with the same constant ν, it follows that Du + V Du ) dx + Du 2 D 2 u 2 dx c A ε u + u ) dx, with c indeendent of ε. Since A ε u Au 0 as ε goes to 0, we get the thesis.

15 Maximal regularity in L ) for a class of ellitic oerators Proof of Theorem 2. In this section we rove Theorem 2., which states that the oerator A, D 2 ) see 2.3)) generates a C 0 -semigrou in L 2 ), which turns out to be a contractive one if c β = 0. The roof goes as follows. As a by-roduct of Lemma 4. we deduce that the a riori estimates roved in Section 4, with = 2 extend to D 2. More recisely, it follows from Lemma 4., Remark 4.3, Lemmas 4.5 and 4.6 that if u D 2 and H), H2), H3), H4), 2.5) and 2.6) hold, then 5.) Du 2 + V u 2 + D 2 u 2 ) dx c Au 2 + u 2 ) dx, for some c deending only on N, ν, α, β, τ, M, Dq ij. By difference, since Au is in L 2 ), then 5.2) F, Du 2 dx c Au 2 + u 2 ) dx, with a ossibly different c. Moreover, A, D 2 ) is closed and densely defined in L 2 ). If c β = 0, then A, D 2 ) is also dissiative. In order to aly the Hille-Yosida Theorem, it remains to rove that λ A : D 2 L 2 ) is bijective for sufficiently large λ and that the corresonding estimate for the resolvent is satisfied. This is roved through a standard rocedure, namely by aroximating the solution of the ellitic equation λu Au = f, f L 2 ), with a sequence of solutions of the same equation in balls with increasing radii and satisfying Dirichlet boundary conditions. Lemma 5.. Suose that H), H2), H3), H4), 2.5) and 2.6) hold. Then A, D 2 ) is closed in L 2 ). Moreover, A c β 2, D 2) is dissiative. Proof. If u D 2, then u A c u D2, A being the grah norm of A, for some ositive c deending on q ij and Dq ij. Moreover, from 5.) and 5.2) there exists c 2 > 0 such that u D2 c 2 u A. This roves that D2 is equivalent to A ; since D 2 is obviously comlete with resect to the former, it turns out that D 2 is also comlete with resect to the latter, which just means that A, D 2 ) is closed. Finally, taking into account Remark 4.4 and Lemma 4., we conclude that A c β 2, D 2) is dissiative. In the roosition below we study the surjectivity of the oerator λ A, for ositive λ. We remark that the injectivity for λ > c β 2 follows from the dissiativity stated in Lemma 5.. Proosition 5.2. Suose that H), H2), H3), H4), 2.5) and 2.6) hold. Then for every f L 2 ) and for every λ > c β /2, there exists a solution u D 2 of 5.3) λu Au = f, in. Moreover, 5.4) u 2 λ c ) β f 2. 2 Proof. We deal with the case c β = 0 only, since the remaining case c β 0 is analogous. For each ρ > 0 consider the Dirichlet roblem λu Au = f, in B ρ 5.5) u = 0, on B ρ,

16 6 G. Cuini - S. Fornaro with λ > 0 and f L 2 ). According to [8, Theorem 9.5] there exists a unique solution u ρ of 5.5) in W 2,2 B ρ ) W,2 0 B ρ ). Let us rove that the dissiativity estimate holds. Multilying λ u ρ L 2 B ρ) f L 2 ) 5.6) λu ρ Au ρ = f by u ρ and integrating by arts with similar estimates as in the roof of Lemma 4.2, taking into account that u ρ = 0 on B ρ, we get λ u 2 ρ dx + ν Du ρ 2 dx div F u 2 ρ dx + V u 2 ρ dx fu ρ dx B ρ B ρ 2 B ρ B ρ B ρ and by H2) it follows λ u 2 ρ dx + ν Du ρ 2 dx + β ) V u 2 ρ dx B ρ B ρ 2 B ρ Then we have ) /2 ) /2 u 2 ρ dx f 2 dx. B ρ B ρ 5.7) u ρ L 2 B ρ) λ f L 2 ), Du ρ L 2 B ρ) ν /2 λ /2 f L 2 ). Multilying 5.6) by V u ρ, with analogous estimates as in the roof of Lemma 4.5 we get the inequality 5.8) V u ρ L 2 B ρ) c f L 2 ), with c indeendent of ρ. Let ρ < ρ 2 < ρ. By [8, Theorem 9.] and 5.7) we obtain ) u ρ W 2,2 B ρ ) c f L 2 B ρ2 ) + u ρ L 2 B ρ2 ) c 2 f L 2 ), with c and c 2 indeendent of ρ. Thus, {u ρ } is bounded in W 2,2 loc RN ), hence there is a sequence {u ρn }, ρ n < ρ n+, weakly convergent to u in W 2,2 loc RN ) and strongly in L 2 loc RN ). Actually, {u ρn } strongly converges to u in W 2,2 loc RN ). In fact, fixed s and t, 0 < s < t, for every n, m such that ρ n, ρ m > t, by [8, Theorem 9.] again, u ρn u ρm W 2,2 B s) cs, t) u ρn u ρm L 2 B t), since both u ρn and u ρm satisfy λu Au = f in B t. The convergence of {u ρn } to u in L 2 B t ) roves that {u ρn } is a Cauchy sequence in W 2,2 B s ) and so the assertion follows. As a consequence, u is a solution of 5.3) for a.e. x. In order to conclude, it remains to rove that u D 2. First, we rove that u W,2 ) and V u L 2 ), then that F, Du L 2 ). Finally, by difference from 5.3) and using classical L 2 -regularity, it follows that u W 2,2 ). By 5.7) and 5.8) we get that, fixed R < ρ n, u 2 ρ n dx u 2 ρ n dx λ 2 f 2 dx, B R B ρn Du ρn 2 dx B R Du ρn 2 dx ν λ B ρn f 2 dx and V u ρn ) 2 dx B R V u ρn ) 2 dx c B ρn f 2 dx.

17 Maximal regularity in L ) for a class of ellitic oerators... 7 Since c does not deend on ρ n and R, letting first n + and then R +, we get 5.4) and Du 2 + V u 2 ) dx c f 2 dx. In articular, u W,2 ) and V u L 2 ). Now, let η Cc ) such that 0 η, η in B, st η B 2 and Dη 2 + D 2 η L. Set η n x) = ηx/n). We have 5.9) Aη n u) η n Au = q ij D j ud i η n + D i q ij ud j η n ) F, Dη n u. Observe that Aη n u) η n Au 0 as n + in the L 2 -norm. In fact, N q ij D j ud i η n + D i q ij ud j η n )) goes to 0 in the L 2 -norm, since u W,2 ) and, arguing as in 4.), we obtain the convergence to 0 for the last term in 5.9), too. Since η n Au Au in L 2, then Aη n u) Au, too. Being η n u D 2, by the equivalence of the two norms D2 and A roved in Lemma 5. we get ) F, Du η n L 2 ) c Aη n u) L 2 ) + η n u L 2 ) + F, Dη n u L 2 ). Letting n +, one then establishes F, Du L 2 ) c ) Au L 2 ) + u L 2 ). By difference, N D i q ij D j u) belongs to L 2 ). Thus, by 2.) and L 2 ellitic regularity the second order derivatives of u are in L 2, which imlies that u W 2,2 ) and u D 2. The roof that the oerator A, D 2 ) generates a strongly continuous semigrou in L 2 ) is now a straightforward consequence of the above results. Proof of Theorem 2.. It is easily seen that A, D 2 ) is densely defined, then the assertion follows from the Hille-Yosida Theorem see [7, Theorem II.3.5]). If c β = 0 then A, D 2 ) is dissiative and therefore the generated semigrou is contractive. 6. Proof of Theorem 2.2 In this section, we rove the generation result Theorem 2.2, which holds for any >. The roof is more involved than that of Theorem 2. given in the revious section, since the variational method fails to estimate the L -norm of the second order derivatives of a solution u D of Au = f, f L ). Thus, we emloy a different technique, which works under more restrictive assumtions on the coefficients of A, recisely we relace assumtions H) and H4) with H ) and H4 ), resectively. As noticed in Section 2, these two assumtions imly 2.8). Moreover, H5) is assumed. The estimate of the second order derivatives is roved in Proosition 6.5. The idea is to define, via a change of variables and a localization argument, a family of oerators, say {A x0 } x0, with a globally Lischitz drift coefficient and a bounded otential term. Then we aly Theorem 3. to each A x0 to obtain local estimates of the L -norm of the second order derivatives of u. In order to get global estimates, we use a covering argument based on Besicovitch s Covering Theorem see Proosition 6. below). We just note that the transformed oerators {A x0 } turn out to be uniformly ellitic if and only if we require that F θv /2, which is the case of [4]. In Proosition 6.7, we deal with the surjectivity of the oerator λ A, the main tool being an aroximation rocedure.

18 8 G. Cuini - S. Fornaro Proosition 6.. Let F = {Bx, ρx))} x be a collection of balls such that 6.) ρx) ρy) L x y, x, y, with L < 2. Then there exist a countable subcovering {Bx n, ρx n ))} and a natural number ζ = ζn, L) such that at most ζ among the doubled balls {Bx n, 2ρx n ))} overla. The above roosition relies on the following version of the Besicovitch covering theorem, see e.g. [, Theorem 2.8]). Proosition 6.2. There exists a natural number ξn) satisfying the following roerty. If Ω is a bounded set and ρ : Ω 0, + ), then there is a set S Ω, at most countable, such that Ω Bx, ρx)) and every oint of belongs at most to ξn) balls Bx, ρx)) centered at x S oints of S. We turn now to the roof of Proosition 6.. Proof of Proosition 6.. If L = 0 then the radii are constant and the statement easily follows. If L > 0, we consider the sets Ω n := B 0, 2ρ0) + L) n) \ B 0, 2ρ0) + L) n ), n Ω 0 := B0, 2ρ0)). Alying Proosition 6.2 we have that for all n N 0 there exists a at most) countable subset S n Ω n, such that Ω n Bx, ρx)) =: C n. Since 6.) imlies ρx) ρ0) + L x, it is easy x S n to rove that ) ) C n B 0, ρ0)2 + L) n+ + ) \ B 0, ρ0)2 L) + L) n ), n. Note that 2 + L) n L) > 0 for all n because L < 2. Since + L >, there exists k = kl) N such that for all n k 2 L) + L) n > 2 + L) n k+ +, which imlies that C n C n k =. Hence the intersection of at most k among the sets C n can be non-emty. Moreover, at most ξn) among the balls centered at oints of S n overla. It turns out that F = {Bx, ρx)) : x S n, n N 0 } =: {Bx j, ρ j )} is a countable subcovering of and if ξ = k ξn) then at most ξ balls of F overla. To estimate the number of overlaing doubled balls {Bx j, 2ρ j )} we roceed as in [4, Lemma 2.2]. Let Bx i, ρ i ) F be fixed and set Ji) = {j N : Bx i, 2ρ i ) Bx j, 2ρ j ) }. If j Ji) it turns out that ρ i ρ j 2Lρ i +ρ j ), because x i x j 2ρ i +ρ j ), yielding 2L +2L ρ i ρ j +2L 2L ρ i. Thus, the balls Bx j, ρ j ), j Ji), are contained in Bx i, 5+2L 2L ρ i). Since at most ξ of the balls Bx j, ρ j ) overla, we obtain 2L + 2L ) N ρ N i card Ji) j Ji) ρ N j ) 5 + 2L N ξ ρ N i, 2L ) which imlies card Ji) ξ 5+2L)+2L) N, 2L) so that the number of overlaing doubled balls is 2 an integer ζ, with ζ + ξ 5+2L)+2L) 2L) 2 ) N. The following simle lemma is a straightforward consequence of assumtion H ) and it will be useful to rove Proosition 6.5 below.

19 Maximal regularity in L ) for a class of ellitic oerators... 9 Lemma 6.3. Assume that H ) holds. Then there exist ε > 0 and two constants a, b > 0, deending on α, σ, µ, such that for all x 0 with av x) V x 0 ) bv x), for every x Bx 0, 3εrx 0 )), 6.2) rx 0 ) := + x 0 2 ) µ/2 V σ x 0 ). Proof. We remark that from the choice of the arameters µ and σ and since V then 6.3) + x 2 ) µ/2 V σ x) + x, for every x. Moreover, H ) is equivalent to one of the following inequalities 6.4) DV σ α σ) x) + x 2 ) µ/2, σ <, α D log V x) + x 2 ) µ/2, σ =. We rove the thesis assuming σ <, the case σ = being analogous. Fix x 0 and write r in lace of rx 0 ). Suose first that x 0 <. From 6.3) and 6.2) it follows that Bx 0, 3εr) B0, 2), for every 0 < ε /6. Moreover, since V is a continuous function and V, we have also that there exist ω, ω 2 > 0, indeendent of x 0, such that ω = inf y B0,2) V y) inf y Bx 0,3εr) V y) V x 0) V x) su V y) = ω 2, y B0,2) x Bx 0, 3εr). Let us now deal with the case x 0. By 6.3) one has ry) + y, y, so that for every 0 < ε /6 su y + y 2 + y 3εr) 2 < +. Therefore, there exist ε /6 and τ both indeendent of x 0, such that 3εα σ) + x 0 2 ) µ/2 + x 0 3εr) 2 ) µ/2 τ <, where α and σ are as in H ). Thus, by the mean value theorem and 6.4) it follows that for every x Bx 0, 3εr) and, multilying by V σ x)v σ x 0 ), V σ x 0 ) τ) V σ x) V σ x 0 ) + τ) 6.5) V σ x) τ) V σ x 0 ) V σ x) + τ). Therefore the statement is roved with a = inf{ω, τ) σ } and b = su{ω2, + τ) σ }. The following algebraic lemma is useful to rove Proosition 6.5. Lemma 6.4. If H ) holds, with σ, µ) 2, 0), then for every δ > 0 there exists c δ > 0 such that 6.6) DV δv 3/2 + c δ.

20 20 G. Cuini - S. Fornaro Proof. If 2 < σ, then 6.6) trivially follows by Young s inequality, with c δ deending only on σ, α and c α. If instead σ = 2, then by assumtion µ > 0. For all δ > 0 choose R δ > 0 such that + x 2 ) µ/2 α/δ for every x \ B Rδ. Hence DV α V 3/2 + x 2 ) µ/2 δv 3/2 + α su x B Rδ V 3/2 x). In the following roosition we extend to the case 2 the estimate of the second order derivatives stated in 5.) in the case = 2. Proosition 6.5. Assume H ), H2 ), H4 ), H5) with constants satisfying 2.7). If u D then 6.7) V u + F, Du + D 2 u ) dx c Au + u ) dx, with c deending only on N,, ν, M, q ij, Dq ij and the constants in H ), H2 ), H4 ) and H5). Proof. By Lemma 4. we may reduce to consider u Cc ). Moreover, for the sake of simlicity and without loss of generality, we can rove the statement assuming c β = 0. Set f = Au. We claim that the assumtions of Lemma 4.5 hold. Since div F N DF then H2 ) imlies 6.8) div F βv with β < because of 2.7). Moreover, H ) and H4 ) imly 2.8), that is F, DV αθv 2. If σ, µ) = 2, 0), then H) trivially follows from H ) and 2.8) imlies 4.9). If instead σ > 2 or µ > 0, then by Lemma 6.4 H) holds, with α and c α relaced by δ and c δ, resectively, with δ arbitrarily small. Choose δ, deending only on N,, M and on the constants in H ), H2 ), H4 ) and H5), such that 6.9) M 4 )δ2 + β + αθ <. Thus, 4.9) holds and Lemma 4.5 imlies 6.0) V u dx c f + u ) dx. It remains to estimate the L -norms of D 2 u and F, Du. We begin by considering the second order derivatives of u. Then, by difference, we obtain the estimate of F, Du. For every x 0, let ε and r = rx 0 ) be as in Lemma 6.3. We oint out that ε is indeendent of x 0. Define y 0 equal to λx 0, with λ := V /2 x 0 ). We consider two cut-off functions η and φ in C c ), 0 η, φ, satisfying the following conditions 6.) η in By 0, ελr), st η By 0, 2ελr), φ in By 0, 2ελr), st φ By 0, 3ελr), Dη 2 + D 2 η + Dφ 2 + D 2 φ L λ 2 r 2,

21 Maximal regularity in L ) for a class of ellitic oerators... 2 for some L > 0, deending on ε, but neither on x 0 nor on y 0. For every x, define y = λx and consider vy) = u y λ). Then v satisfies the equation D yi q ij D yj v)y) λ F y), D y vy) λ 2 Ṽ y)vy) = λ fy), 2 y with q ij y) = q y ij λ), F y) = F y ) λ, Ṽ y) = V y ) λ and fy) = f y λ ). Setting wy) = ηy)vy) we deduce that 6.2) with g defined as follows 6.3) D yi q ij y)d yj wy)) λ F y), D y wy) Ṽ y)wy) = gy) λ2 gy) := λ 2 ηy) fy) + 2 qy)dηy), Dvy) + div qdη)y)vy) λ F y), Dηy) vy), y. Since st w By 0, 2ελr), equation 6.2) is equivalent to D yi q ij y)d yj wy)) λ φy) F y), D y wy) λ 2 φy)ṽ y)wy) = gy), y RN. Now, let us define the oerator 6.4) Ã = D yi q ij D yj ) λ φ F, D y λ 2 φ Ṽ. Claim. φ λ Ṽ and 2 λ φ F, D q ij are bounded in and λ φ F is globally Lischitz in with φ λ Ṽ 2, λ φ F, D q ij and the Lischitz constant of λ φ F indeendent of x 0. Proof of claim. The main tool is Lemma 6.3. Recalling the definition of λ, Ṽ and the relationshi between y and x, from Lemma 6.3 it follows that su φy) Ṽ y) y λ2 su x Bx 0,3εr) V x) V x 0 ) a, Taking into account assumtions H2 ), H4 ) and 6.), we have that λ D ) y φy) F y) = su y λ 2 D xf ) λ su y y By 0,3ελr) βv x) su x Bx 0,3εr) V x 0 ) + L β su x Bx 0,3εr) V x) V x 0 ) + Lθ ) φy) + λ F y λ su x Bx 0,3εr) su x Bx 0,3εr) F x) r V x 0 ) Using Lemma 6.3 and equation 6.3) we infer that λ D ) y φy) F y) β a + Lθ [ + x 0 + 3εr) 2 ] a σ + x 0 2 ) µ 2 su y β a + L θ 8 µ 2 a σ ) D y φy) + x 2 ) µ 2 V σ x) + x 0 2 ) µ 2 V σ x 0 ) µ 2

22 22 G. Cuini - S. Fornaro which imlies that λ φ F is globally Lischitz in, uniformly with resect to x 0. Finally, assumtion H5) yields λ φy) F y), D y q ij y) su λ F y), D y q ij y) su y because of Lemma 6.3 and V. y By 0,3ελr) su x Bx 0,3εr) κ su x Bx 0,3εr) λ 2 F x), Dq ijx) V x) V x 0 ) + c κ su x Bx 0,3εr) V x 0 ) κ a + c κ, Claim 2. The function g in 6.3) satisfies the estimate 6.5) gy) dy C λ 2 N ux) + fx) + V x)ux) + V /2 x)dux) ) dx, Bx 0,2εr) for some C deending on ε, but not on x 0. Proof of claim 2. We searately consider each term of g. The constants occurring in the estimates may deend on ε. The first term in 6.3) is the easiest to estimate, in fact 6.6) ) y λ 2 ηy)f λ dy λ 2 By 0,2ελr) f y λ ) dy = λ 2 N fx) dx. Bx 0,2εr) Using 6.) we can estimate the L -norm of the next two terms as follows 2 qy)d y ηy), D y vy) dy C ) λ 2 r y Du dy By 0,2ελr) λ C = λ 2 N r Dux) dx = C V σ) x 0 ) λ 2 N + x 0 2 ) µ/2 Dux) dx and Bx 0,2εr) div qdη)y)vy) dy C 2 λ 2 r 2 C 2 = λ 2 N r 2 ux) dx = C 2 λ 2 N Bx 0,2εr) Bx 0,2εr) By 0,2ελr) Bx 0,2εr) vy) dy V 2 σ) x 0 ) + x 0 2 ) µ ux) dx, with C and C 2 indeendent of x 0. Recalling that V, σ 2, µ 0 and using Lemma 6.3, we obtain V σ) x 0 ) Bx 0,2εr) + x 0 2 ) µ/2 Dux) dx V /2 x 0 )Dux) dx Bx 0,2εr) b /2 V /2 x)dux) dx and Bx 0,2εr) Bx 0,2εr) V 2 σ) x 0 ) + x 0 2 ) µ ux) dx V x 0 )ux) dx b V x)ux) dx. Bx 0,2εr) Bx 0,2εr)

23 Maximal regularity in L ) for a class of ellitic oerators Hence, there exists C 3 indeendent of x 0 such that the following inequality holds 6.7) 2 qy)d y ηy), D y vy) + div qdη)y)vy) ) dy C 3 λ 2 N Bx 0,2εr) V x)ux) + V /2 x)dux) ) dx. Concerning the last term in 6.3), we use again assumtion H4 ) and we get λ F y), Dηy) vy) dy c F x) ux) 6.8) λ 2 N Bx 0,2εr) r dx c θ + x 2 ) µ/2 V σ x) λ 2 N Bx 0,2εr) + x 0 2 ) µ/2 V σ V x)ux) dx x 0 ) C 4 λ 2 N V x)ux) dx Bx 0,2εr) where C 4 is not deending on x 0. Thus, the claim is roved since collecting 6.6) 6.8), inequality 6.5) follows. Let us now rove 6.7). Alying Theorem 3. with B relaced by Ã, we have D 2 wy) dy K wy) + gy) ) dy, with K indeendent of x 0. By the definition of w it follows that D 2 vy) dy K vy) + gy) ) dy By 0,ελr) By 0,2ελr) and consequently, since y = λx, λ 2 N D 2 u dx Bx 0,εr) K λ N u dx + K Bx 0,2εr) λ 2 N u + f + V u + V /2 Du ) dx. Bx 0,2εr) Multilying both sides of the revious inequality by λ 2 N and recalling that λ = V /2 x 0 ) we obtain D 2 u dx Bx 0,εr) K which imlies 6.9) Bx 0,2εr) Bx 0,εr) V x 0 )ux) dx + K D 2 u dx K 2 Bx 0,2εr) Bx 0,2εr) u + f + V u + V /2 Du ) dx, u + f + V u + V /2 Du ) dx, because of Lemma 6.3. Now, in order to aly Proosition 6. we need to verify the Lischitz continuity of the radius ε r with resect to x 0. To this aim, we remark that from assumtion H ) it follows that Dεr)x) = ε µ + x 2 ) µ 2 xv σ x) + σ ) + x 2 ) µ 2 V σ 2 x)dv x) { } ε + x 2 ) µ 2 V σ x) + σ) + x 2 ) µ 2 V σ 2 x) DV x) ε { + σ)α}

24 24 G. Cuini - S. Fornaro which is less than /2, choosing a smaller ε if necessary. Let {Bx j, εr j )} be the covering of yielded by Proosition 6.. Alying 6.9) to each x j and summing over j, it follows that D 2 u dx D 2 u dx j N Bx j,εr j ) K 2 u + f + V u + V /2 Du ) dx = K 2 j N ζ K 2 Bx j,2εr j ) ux) + fx) + V x)ux) + V /2 x)dux) ) u + f + V u + V /2 Du ) dx, j N χ Bxj,2εr j )x) dx where ζ is given by Proosition 6.. Thus, using the interolatory estimate [4, Proosition 2.3] and taking into account 6.0) it turns out that D 2 u dx c f + u ) dx, for some c > 0 deending on the stated quantities. Once that the estimate of the second order derivatives is available, by difference we get the estimate for F, Du, that is F, Du dx c f + u ) dx. As in Lemma 5. we can rove that D and A are equivalent norms. This easily imlies the closedness of A, D ). Lemma 6.6. Suose that H ), H2 ), H4 ) and H5) hold, with constants satisfying 2.7). Then A, D ) is closed in L ). Moreover, A c β, D ) is dissiative. The following result deals with the bijectivity of the oerator λ A. It is analogous to Proosition 5.2, but the roof is different. Here we clarify the reason why we require assumtion 2.7), which is stronger than the corresonding one for = 2. In fact, also the oerators A ε defined below must satisfy our hyotheses. Proosition 6.7. Suose that H ), H2 ), H4 ) and H5) hold, with constants satisfying 2.7). Then for every f L ) and for every λ > c β a unique solution u D of λu Au = f, in exists. Moreover, 6.20) u λ c ) β f. Proof. Uniqueness and estimate 6.20) immediately follow from 4.7). As far as the existence is concerned, for fixed ε > 0, let us define F ε : and V ε : R as F ε := F + εv, V ε := V + εv. It is easy to rove that H ), H2 ), H4 ) and H5) imly H ε ) DV ε x) α V 2 σ ε x) + x 2 ) µ/2,

25 Maximal regularity in L ) for a class of ellitic oerators H ε 2) DF ε 2 β N + αθ)v ε + 2 N c β, H ε 4) F ε x) θ + x 2 ) µ/2 V σ ε x), H ε 5) F ε x), Dq ij x) κ V ε x) + c κ, resectively. Assumtions H ε ), H ε 2) and H ε 4) yield 6.2) div F ε 2β + Nαθ)V ε + 2 c β, F ε, DV ε αθv 2 ε and D i Fε j x)ξ i ξ j ) β 2 2 N + αθ V ε x) ξ 2 N c β ξ 2, ξ, x. Notice that V ε is bounded and F ε is globally Lischitz in. Precisely, V ε ε, and D ifε j ) β N + αθ + c β, i, j N. ε N ) Moreover, if σ, µ) 2, 0 arguing as in the roof of Lemma 6.4 and observing that V ε V, we have that for every δ > 0 there exists c δ 0 such that 6.22) DV ε δvε 3/2 + c δ, for every ε > 0. Therefore, the above inequality and 2.7) imly that there exists δ > 0 indeendent of ε such that M 6.23) 4 )δ2 + 2 β + Nαθ + αθ <. Let us consider the oerator A ε := A 0 F ε, D V ε where, as reviously defined, A 0 stands for N D i q ij D j ). Define D,ε and its norms D,ε and Aε analogously to D, D and A, resectively, that is { } D,ε := u W 2, ) : F ε, Du L ), u D,ε := u 2, + V ε u + F ε, Du, u Aε := A ε u + u. Since the constants involved in H ε ), H ε 2), H ε 4), H ε 5) and 6.23) are indeendent of ε, from Lemma 6.6 we get that there exist k and k 2, indeendent of ε, such that 6.24) k u Aε u D,ε k 2 u Aε. Since the oerator A ε satisfies the assumtions of Proosition 3.2, for every λ > 2 c β one has λ ρa ε ) and Rλ, A ε ) λ 2 c ) β. In fact, using the inequality Vε + ε), the first estimate in 6.2) and noting that 2.7) imlies 2 β+ Nαθ <, we get { su x divf εx) V ε x)} 2 β + ) Nαθ + 2 c β + ε < 2 c β. Therefore, if λ > 2 c β then for every f L ) and for all ε > 0, there exists a unique u ε D,ε such that 6.25) λu ε A ε u ε = f, in

26 26 G. Cuini - S. Fornaro and 6.26) u ε λ 2 c ) β f. Using 6.24), 6.25) and 6.26) we obtain 6.27) u ε D,ε k 2 A ε u ε + u ε ) k 2 + λ + ) λ 2 c f β. In articular, we have that {u ε } is bounded in W 2, ), thus there exist u W 2, ) and a sequence {u εn } converging to u weakly in W 2, ) and strongly in W, loc RN ). Therefore, u to a subsequence, u εn u and Du εn Du a.e. in. From 6.27) we obtain in articular that V εn u εn + F εn, Du εn c f, which imlies, using Fatou s Lemma, that V u + F, Du c f. Thus, u D. It remains to rove that u solves λu Au = f a.e. in. From 6.25) and the definition of A εn we infer that λu εn A 0 u εn = f εn, where f εn = f F εn, Du εn V εn u εn L ). Alying the classical local L -estimates see [8, Theorem 9.]) it follows that for every 0 < ρ < ρ ) u εn W 2, B ρ ) C f εn L B ρ2 ) + u εn L B ρ2 )), with C deending on ρ, ρ 2 but indeendent of n. Since u εn and f εn converge to u and f F, Du V u, resectively, in L loc RN ) as n, by alying 6.28) to the difference u εn u εm we get that {u εn } is a Cauchy sequence in W 2, B ρ ). This imlies that u εn converges to u in W 2, loc RN ) and then, letting n in the equation solved by u εn, it follows that u satisfies λu Au = f a.e. in. To conclude the roof it remains to show that λ A is surjective also when λ > c β. This follows from the dissiativity of the oerator A c β, stated in Lemma 6.6, and the fact that λ A c β is surjective for λ > 2 )c β /. Thus λ A c β λ A is surjective for λ > c β, as claimed. We are ready to rove Theorem 2.2. ) ) is also surjective for λ > 0, which means that Proof of Theorem 2.2. Since Cc ) D L ), it follows that D is a dense subset in L ). Moreover, A, D ) is closed, by Lemma 6.6. By Proosition 6.7 and 6.20), for every λ > c β, λ A : D L ) is bijective and λ A) f λ c ) β f. The thesis follows from the Hille-Yosida Theorem.

27 Maximal regularity in L ) for a class of ellitic oerators Comments and consequences In this final section we establish some further roerties of the semigrou T ) generated by A, D ) on L ). We note that since all the assumtions of Theorem 2.2 for = 2 imly those of Theorem 2., the semigrou T 2 ) is uniquely determined. We oint out that the semigrous given by Theorem 2.2 are not analytic, in general. An examle is the Ornstein-Uhlenbeck semigrou see e.g. [, Examle 4.4]). In the following roosition we rove the consistency of T ). Proosition 7.. Assume that the assumtions of Theorem 2.2 hold for some and q, with <, q < +. If f L ) L q ) then T t)f = T q t)f, for all t 0. Proof. By [7, Corollary III.5.5] we have only to rove that the resolvent oerators of A, D ), A, D q ) are consistent, for λ large, i.e. that for every f L ) L q ) there exists u W 2, ) W 2,q ) such that λu Au = f. This follows from the roofs of Proosition 6.7 and [3, Theorem 2.2] since the same roerty holds for uniformly ellitic oerators. Now we rove the ositivity of T. Proosition 7.2. T ) is ositive, i.e. if f L ), f 0, then T t)f 0, for all t 0. Proof. The ositivity of the semigrou T is equivalent to the ositivity of the resolvent λ A) for all λ sufficiently large. By the roof of Proosition 6.7 this last roerty turns out to be true once that each A ε is shown to have a ositive resolvent. From [3, Theorem 2.2] this holds because the oerators A ε can be aroximated by uniformly ellitic oerators. In the following roosition we show the comactness of the resolvent of A, D ) assuming that the otential V tends to infinity as x +. This result is similar to [4, Proosition 6.4] and we give the roof for the sake of comleteness. Proosition 7.3. If lim x + V x) = + then the resolvent of A, D ) is comact. Proof. Let us rove that D is comactly embedded into L ). Let F be a bounded subset of D. Then, by the assumtion, given ε > 0 there exists R > 0 such that 7.) fx) dx ε x >R for every f F. Since the embedding of W 2, B R ) into L B R ) is comact, the set F = {f BR f F}, which is bounded in W 2, B R ), is totally bounded in L B R ). Therefore there exist r N and g,..., g r L B R ) such that r 7.2) F {g L B R ) g g i L B R ) < ε}. Set i= g i = { gi in B R 0 in \ B R. Then g i L ) and from 7.) and 7.2) it follows that r F {g L ) g g i < 2ε}. i= This imlies that F is relatively comact in L ) and the roof is comlete.