ABSTRACT
The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in (12) which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.
Keywords:
exact boundary controllability; non-cylindrical domains; linear Klein-Gordon equation
RESUMO
O objetivo deste artigo é estudar um problema de controlabilidade exata na fronteira em domínios não cilíndricos para a equação linear de Klein-Gordon. Aqui, trabalhamos pro´ximo das técnicas de extensão apresentadas por J. Lagnese, em (12), que é baseada no método de controlabilidade de Russell. O tempo de controle é obtido em qualquer instante maior que o valor do diâmetro do domínio no qual os dados iniciais estão suportados. O controle é de quadrado integrável e atua em toda fronteira e é obtido por meio da derivada conormal associada ao operador linear de Klein-Gordon.
Palavras-chave:
controle exato na fronteira; dominios não-cilindricos; equação linear Klein-Gordon
1 INTRODUCTION
Since the second half of the last century to now we have seen an increase in the interest of the mathematicians and engineers in the study of vibratory problems modeled by the wave equation. This can be confirmed by great volume of the works dealing with such problems presented in the literature. In fact, such problems have theoretical an practical importance because have wide applications in many branches of the engineering and mathematics.
An interesting part is to study vibratory problems that occur on a flexible body whose boundary deforms over time. Such phenomena generate, on the space-time scale, the non-cylindrical domains. An illustrative example is a metallic body in vibration that is placed in an environment subject to a change in the temperature. The increasing or decreasing of the temperature causes a linear expansion or contraction of the metallic body characterizing a movement of its boundary along of the time generating a non-cylindrical domain.
In the literature we can find many works that study the most diverse types of phenomena involving wave equations over non-cylindrical domains. Problems involving the existence of solutions, energy decay, stabilization and control of wave equations on non-cylindrical domains have already been considered by several authors, to cite a few, see 11 N.L. Balazs. On the solution of the wave equation with moving boundaries. J. Math. Anal. Appl., 3 (1961), 472-484.),(22 C. Bardos & G. Chen. Control and stabilization for wave equation, part III: domain with moving boundary. SIAM J. Control Optim., 19 (1981), 123-138.),(33 C. Bardos , G. Lebeau & J. Rauch . Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim ., 30 (1992), 1024-1065.),(44 W.D. Bastos & J. Ferreira. Exact boundary control for the wave equation in a polyhedral time-dependent domain. App. Math. Lett., 12 (1999), 1-5.),(55 J. Cooper. Local decay of solutions of the wave equation in the exterior of a moving body. J. Math. Anal. Appl. , 49 (1975), 130-153.),(66 J. Cooper & C. Bardos. A nonlinear wave equation in a time-dependent domain. J. Math. Anal. Appl. , 42 (1973), 29-60.),(77 L. Cui. The wave equation with internal control in non-cylindrical domains. Adv. Differ. Equ., 2017 (2017), 267.),(88 L. Cui , X. Liu & H. Gao . Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. , 402 (2013), 612-625.),(1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.),(1313 J.L. Lions. Exact controllability, stabilization and perturbations for distributed system. SIAM Rev., 30 (1988), 1-68.),(1414 L. Lu , S. Li , G. Chen & P. Yao . Control and stabilization for the wave equation with variable coefficients in domains with moving boundary. Systems Control Lett., 80 (2015), 30-41.),(1515 M.M. Miranda. Exact controllability for the wave equation in domains with variable boundary. Rev.Mat. Univ. Complut. Madrid, 9 (1996), 435-457.),(1818 D.L. Russell. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math., 52 (1973), 189-211.. Especially the problems that deal with control processes for wave equations have been extensively studied in the last 50 years. We can find in the literature many works that brought a great contribution to the development of the control theory for hyperbolic equations. In highlight we can cite the papers ( 33 C. Bardos , G. Lebeau & J. Rauch . Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim ., 30 (1992), 1024-1065.), (1313 J.L. Lions. Exact controllability, stabilization and perturbations for distributed system. SIAM Rev., 30 (1988), 1-68.), (1818 D.L. Russell. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math., 52 (1973), 189-211.) which established important methods for the study of controllability of wave equations, but they have worked only on cylindrical domains.
When we look specifically at control problems for waves equations in non-cylindrical domains we realize that, in the literature, the number of these is very small when compared to control problems for waves in cylindrical domains. But this seems natural, since the work on cylindrical domains is simpler than on non-cylindrical domains and often to study control problems in non-cylindrical domains we need, at some point, to resort to the techniques developed to treat controllability problems on the cylindrical domains. However, we can find interesting papers in the literature that deal with exact controllability for wave equations in non-cylindrical domains with the most diverse types of controllability methods, to cite a few see 22 C. Bardos & G. Chen. Control and stabilization for wave equation, part III: domain with moving boundary. SIAM J. Control Optim., 19 (1981), 123-138.), (44 W.D. Bastos & J. Ferreira. Exact boundary control for the wave equation in a polyhedral time-dependent domain. App. Math. Lett., 12 (1999), 1-5.), (77 L. Cui. The wave equation with internal control in non-cylindrical domains. Adv. Differ. Equ., 2017 (2017), 267.), (88 L. Cui , X. Liu & H. Gao . Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. , 402 (2013), 612-625.), (1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.), (1414 L. Lu , S. Li , G. Chen & P. Yao . Control and stabilization for the wave equation with variable coefficients in domains with moving boundary. Systems Control Lett., 80 (2015), 30-41.), (1515 M.M. Miranda. Exact controllability for the wave equation in domains with variable boundary. Rev.Mat. Univ. Complut. Madrid, 9 (1996), 435-457. and citing papers.
Taking into account the importance of the theme, this work proposes to study, in the light of the ideas of 1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135., an exact boundary controllability problem for the linear Klein-Gordon equation in non-cylindrical domains. In order to establish the results let us consider , be a bounded and smooth by parts domain whose value of the its diameter will be denoted by d(Ω), and c a positive real number. Let Q be a non-cylindrical open set in such that the intersection of Q with hyperplane is a nonempty bounded open set Ωt in ℝN and such that . We represent the boundary of Ωt by ∂Ωt and is a variety that inherits from Ω the property of the smoothness by parts. Now, for , we set
In order to guarantee the well-posedness of the initial and boundary value problems to be considered we requires that Q T , for , be contained in the conical time-like region
Being an arbitrary domain, we denote by L 2(U) and H 1(U) the Lebesgue and Sobolev spaces, provided with theirs usual norms which will be denoted by and respectively (see 99 R. Dautray & J.L. Lions. “Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2”. Cambridge University Press, Berlin (1988).). Let us also consider the space which is the closure of in H 1(U) provided with the norm of H 1(U). The product space is considered endowed with the product norm . The principals results of this paper are established in the two below theorems.
Theorem 1.1.Letandbe fixed. For every, with exception of at most a finite of number of them, there is a extensionsuch that the solution of the Cauchy problem
vanishes in the finite cylinder, that is
The next theorem shows how the Theorem 1.1 can be used to obtain the exact boundary value control of the solutions of the linear Klein-Gordon equation in certain non-cylindrical domains.
Theorem 1.2. Let andbe such that. Then there is a control functionsuch that the solution of
satisfies the final condition
Here (ν x , ν t ) denotes the outward unit normal vector on the surface ΓT at (x,t) point. The expression denotes the conormal derivative of u on ΓT at point (x,t). If Q T were a cylindrical domain we have , so the control function h would be determined by normal derivative of u. In (1.7) one has a time dependent boundary condition. Such conditions is very important in mathematical physics when dealing with diffraction problems for wave equations that need to be limited in a region of the space and some wave signs in the boundary acquire a velocity normal to its surface (see 1010 F.G. Friedlander. “Sound Pulses”. Cambridge University Press, Cambridge (1958). ).
We have mentioned above some papers that deal with different types of control problems for wave equations in non-cylindrical domains which use different controllability methods present in the literature. With respect the types of controllability problems the papers ( 1414 L. Lu , S. Li , G. Chen & P. Yao . Control and stabilization for the wave equation with variable coefficients in domains with moving boundary. Systems Control Lett., 80 (2015), 30-41.), (99 R. Dautray & J.L. Lions. “Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2”. Cambridge University Press, Berlin (1988).) (77 L. Cui. The wave equation with internal control in non-cylindrical domains. Adv. Differ. Equ., 2017 (2017), 267.), (22 C. Bardos & G. Chen. Control and stabilization for wave equation, part III: domain with moving boundary. SIAM J. Control Optim., 19 (1981), 123-138.) have worked about internal exact controllability while the papers (1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.), (88 L. Cui , X. Liu & H. Gao . Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. , 402 (2013), 612-625.), (1515 M.M. Miranda. Exact controllability for the wave equation in domains with variable boundary. Rev.Mat. Univ. Complut. Madrid, 9 (1996), 435-457.), (44 W.D. Bastos & J. Ferreira. Exact boundary control for the wave equation in a polyhedral time-dependent domain. App. Math. Lett., 12 (1999), 1-5.) deal with exact boundary controllability. With respect the methods of controllability the references (77 L. Cui. The wave equation with internal control in non-cylindrical domains. Adv. Differ. Equ., 2017 (2017), 267., 88 L. Cui , X. Liu & H. Gao . Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. , 402 (2013), 612-625., 1515 M.M. Miranda. Exact controllability for the wave equation in domains with variable boundary. Rev.Mat. Univ. Complut. Madrid, 9 (1996), 435-457.) have used the HUM Method establish in 1313 J.L. Lions. Exact controllability, stabilization and perturbations for distributed system. SIAM Rev., 30 (1988), 1-68. for obtain its desirable controllability. The papers (1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.), (44 W.D. Bastos & J. Ferreira. Exact boundary control for the wave equation in a polyhedral time-dependent domain. App. Math. Lett., 12 (1999), 1-5.) have worked near the Russell’s controllability method which is established in 1818 D.L. Russell. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math., 52 (1973), 189-211.. By the end in (1414 L. Lu , S. Li , G. Chen & P. Yao . Control and stabilization for the wave equation with variable coefficients in domains with moving boundary. Systems Control Lett., 80 (2015), 30-41.), (22 C. Bardos & G. Chen. Control and stabilization for wave equation, part III: domain with moving boundary. SIAM J. Control Optim., 19 (1981), 123-138.) the authors obtain controllability by means of techniques based in terms of the Riemmannian metrics and geometric multipliers.
In the present paper we work in the light of Russell’s controllability method in order to prove the Theorem 1.2 and also extension techniques showed in 1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135. in order to proof the Theorem 1.1. In the reference 1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135. the author used a extension technique, based in the Russell’s controllability method, to obtain control for the standard wave equation in non cylindrical domains. As far as we know, there is no work in the literature that studies control for the Klein-Gordon equation in non-cylindrical domains using the methods as presented here. So, this is the contribution of this work. Finally, it is opportune to highlight the importance and usefulness of this method, because it is more easy to handle when dealing with domains that do not have a smooth geometry as proposed here, and this would be a difficulty to deal, for example, with the HUM method.
The rest of the present paper is organized in the following manner. The Section 2 is dedicated to do some essential preliminaries results. The Section 3 is devoted to prove the Theorem 1.2. The Theorem 1.1 is proved in the Section 4 and the paper is finalized with a references section.
2 SOME PRELIMINARIES
In this section we will highlight three important results that are essential in the proofs of the theorems proposed in the previous section. Such results are local energy decay estimates, analytic extension and suitable trace theorem to measure the regularity of the conditional derivative along of surfaces for the solution of the Cauchy problem to the Klein-Gordon equation.
2.1 Energy decay and analyticity
Let be a bounded domain and initial data supported in U. Considering the Cauchy problem
It is known that the unique solution u of Cauchy problem (2.1)-(2.2) is such that and it was established in 1616 R.S.O. Nunes & W.D. Bastos. Energy decay for the linear Klein-Gordon equation and boundary control. J. Math. Anal. Appl. , 414 (2014), 934-944. explicit formulas for the solution u when .
Let be the solution of (2.1)-(2.2), of each we define the operator solution by . The operator S t applies the initial state into final state and for such operator is compact, bounded and liner. In the proof of the Theorem 1.1 will be crucial the following result.
Lemma 2.1.Let u be the solution of the Cauchy problem (2.1)-(2.2) with initial datacompactly supported in U and ℜ the restriction operator to U. There exist positive real constantsand K, independent of u 0 and u 1 such that
for every.
The proof of the above lemma is obtained by a direct manipulation of the Theorem 2.1 of 1616 R.S.O. Nunes & W.D. Bastos. Energy decay for the linear Klein-Gordon equation and boundary control. J. Math. Anal. Appl. , 414 (2014), 934-944..
Considering still as the solution of the Cauchy problem (2.1)-(2.2) with initial state compactly supported in U , another essential element for the proof of the Theorem 1.1 is the analytic extension of the map to the sector as proved in 1717 R.S.O. Nunes & W.D. Bastos. Analyticity and near optimal time boundary controllability for the linear Klein-Gordon equation. J. Math. Anal. Appl. , 445 (2017), 394-406.. For our purpose, such result can be adapted in terms of the operators St in the next lemma.
Lemma 2.2.Letbe the solution of Cauchy problem (2.1)-(2.2) with initial statecompactly supported in U and considers the operators Stas defined above. Then the family of compact linear operatorextends analytically to a family of linear compact operators, where Σ0 is complex sector, being T 1 any constant greater than d(Ω). The proof of the Lemma 2.2 is an immediate consequence of the Theorem 1.1 of 1717 R.S.O. Nunes & W.D. Bastos. Analyticity and near optimal time boundary controllability for the linear Klein-Gordon equation. J. Math. Anal. Appl. , 445 (2017), 394-406. rewritten it in terms of operators S t .
2.2 Trace regularity
In this part we express a result on the regularity of the traces of the solution of the Klein-Gordon which it is essential to proof the Theorem 1.2. Let us begin take account some notations and definitions. Let be a linear second order hyperbolic partial differential equation with C ∞ coefficients depending on ξ in some open bounded domain . Being an oriented smooth hypersurface which is time-like and non-characteristic with respect to . Let be a unit normal to Σ. If is the principal part of , then the expression defines the conormal derivative of u relative to the along Σ. An important fact it is to know what the regularity of the traces of the conormal derivative on surfaces, for this purpose we turn to the paper 1919 D. Tataru. On regularity of the boundary traces for the wave. Ann. Scuola Norm. Pisa, C. L. Sci., 26 (1998), 185-206.. Considering , with , the Theorem 2 of 1919 D. Tataru. On regularity of the boundary traces for the wave. Ann. Scuola Norm. Pisa, C. L. Sci., 26 (1998), 185-206. proves that if is such that then . Particularly, if we consider as being the Klein-Gordon operator, so its principal part will be . Now, if γ is a smooth hypersurface in ℝN let us consider the surface whose the unit normal vector is where . In this case the conormal derivative of u along is . Particularly, if we apply the trace result mentioned in the previous paragraph for the Klein-Gordon operator we obtain the following result.
Lemma 2.3.Let u be the solution of the Cauchy problem (2.1)-(2.2) with initial data. Let γ be a smooth hypersurface in ℝN , with, with no self intersection and considers the surfacewhich the unit normal vector is. Then the conormal derivative of u alonghas trace.
3 PROOF OF THE THEOREM 1.2
Choose and such that . Given , take for which exists an extension according Theorem 1.1, and let be the solution to the Cauchy problem (1.2)-(1.3) with initial data Note that the solution satisfy. in Ω. Now, in order to obtaining the desired control function h we use the trace result available in the previous section. As we have considered the space dimension we can use the Lemma 2.3 to conclude that the trace of the conormal derivative of is locally square integrable along of the surface Γ, that is . Hence . To finish the prove we define the restriction of to the domain and note that . Now we extend to a function u defined on Q T by setting . After defines on ΓT . Note that and satisfy in ΩT see that u and h satisfy the conditions of the Theorem 1.2.
4 PROOF OF THE THEOREM 1.1
We begin the proof with a similar construction that one made by Russell in 1818 D.L. Russell. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math., 52 (1973), 189-211.. Let and Ωδ a δ-neighborhood of the domain Ω. Let be a bounded linear operator that takes the pair to the pair , where and are extension of w 0 and w 1 to ℝN respectively, both with compact support in Ωδ . Let be the solution of the Cauchy problem
Being w the Solution of the Cauchy Problem (4.1)- (4.2) , and taking account the operator as defined in the Section 2. By the Lemma 2.1, with , we have the existence of positive real constants and K, independent of and such that is valid the estimate
for every and is the restriction operator to Ωδ .
Now, let be a cut off function such that . Note that . Following, we solve the Cauchy problem
and we define the operator by being z the solution of the Cauchy problem (4.4)-(4.5). It is important to highlight the relationship between the operators and S T as follows: Let P i be the projection of onto . Then
So, from relations above we can guarantee that the operators have the same properties that the operators S T . So, for and the validity of the inequality
for every , being T 0 and K as in the inequality (4.3).
Taken w and z the solutions of the Cauchy problems (4.1)-(4.2) and (4.4)-(4.5) respectively, we define . Note that the function u satisfy
Observe that by definition of u we have
Keep in mind (4.9) and (4.10), in order of u to be the solution of the Cauchy problem (1.2)-(1.3) satisfying the condition (1.4) it is necessary only guarantee that are extension of f and g respectively from Ω to all ℝN . If this is possible then be the desirable extension of f and g respectively. That is equivalent to guarantee the solution the existence of solution of the equations
In terms of the operators S T and the equations above are equivalents to the equation
Inserting the operator restriction to Ω, denoted by ℜ, the equation (4.11) becomes
being is the variable of such equation taken on space . For any , K T is a linear compact operator on the space . Now, to show that equation (4.12) has solution is equivalent to show the existence of inverse operator , for this purpose it is sufficient to guarantee the contractivity of the family of operators . It is in this point where the inequalities (4.3) and (4.8) will take place. For follows that
where K and C are constants independent from w 0 and w 1. So, from the sequences of inequalities above we obtain, for
where is a constant dependent only K and C. Thus, by inequality (4.13) we guarantee that K T is contraction for a great sufficiently. This ensures the existence of great sufficiently for which the equation (4.12) has solution. However, we would like to establish a lower bound for values of T for which (4.12) is invertible. It is in this point that analytic extension expressed in the Lemma 2.2 will be useful. Because from relationships (4.6) and (4.7) the family of operators have the same properties of the family of operators , so the analytic extension expressed in the Lemma 2.2 is also valid for the family of operators . Thus, we have the guarantee that the family of linear compact operators can be extended analytically to a family of linear compact operators , where Σ0 is complex sector , being T 1 any constant greater than d(Ω). After, we utilize the theorem of alternative of F. V. Atkinson (see 1111 T. Kato. “Perturbation theory for linear operators”. Springer-Verlag, New York (1966).) p. 370) to the effect either 1 is eigenvalue of each of the operators K ζ , or else exists for all except at most for a finite number of values of ζ in each compact subset of Σ0. As observed, for a real sufficiently large K ζ is a contraction, i.e., 1 is not eigenvalue of K ζ , hence the later possibility must be the case. So, for all with the possible exception of a finite number of values, exists, concluding the proof of the Theorem 1.1.
Remark 1.Letbe the midpoint of a internal segment of Ω of length d(Ω). Forletdenotes the open ball with center inand radius r. Considering the infinite cone
We suspect that there may be a extensionofsuch that the solution of (1.2)-(1.3) with Cauchy datavanishes in the finite cone of every . The confirmation of this fact allow us to obtain control for the linear Klein-Gordon equation in some non-cylindrical domains, with control time for everysatisfying. Similar result was obtained in1212 J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.for even dimensional wave equation for. We intend to soon return to this subject.
REFERENCES
-
1N.L. Balazs. On the solution of the wave equation with moving boundaries. J. Math. Anal. Appl., 3 (1961), 472-484.
-
2C. Bardos & G. Chen. Control and stabilization for wave equation, part III: domain with moving boundary. SIAM J. Control Optim., 19 (1981), 123-138.
-
3C. Bardos , G. Lebeau & J. Rauch . Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim ., 30 (1992), 1024-1065.
-
4W.D. Bastos & J. Ferreira. Exact boundary control for the wave equation in a polyhedral time-dependent domain. App. Math. Lett., 12 (1999), 1-5.
-
5J. Cooper. Local decay of solutions of the wave equation in the exterior of a moving body. J. Math. Anal. Appl. , 49 (1975), 130-153.
-
6J. Cooper & C. Bardos. A nonlinear wave equation in a time-dependent domain. J. Math. Anal. Appl. , 42 (1973), 29-60.
-
7L. Cui. The wave equation with internal control in non-cylindrical domains. Adv. Differ. Equ., 2017 (2017), 267.
-
8L. Cui , X. Liu & H. Gao . Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. , 402 (2013), 612-625.
-
9R. Dautray & J.L. Lions. “Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2”. Cambridge University Press, Berlin (1988).
-
10F.G. Friedlander. “Sound Pulses”. Cambridge University Press, Cambridge (1958).
-
11T. Kato. “Perturbation theory for linear operators”. Springer-Verlag, New York (1966).
-
12J. Lagnese. On the support of solutions of the wave equation with applications to exact boundary value controllability. J. Math. pures et appl., 58 (1979), 121-135.
-
13J.L. Lions. Exact controllability, stabilization and perturbations for distributed system. SIAM Rev., 30 (1988), 1-68.
-
14L. Lu , S. Li , G. Chen & P. Yao . Control and stabilization for the wave equation with variable coefficients in domains with moving boundary. Systems Control Lett., 80 (2015), 30-41.
-
15M.M. Miranda. Exact controllability for the wave equation in domains with variable boundary. Rev.Mat. Univ. Complut. Madrid, 9 (1996), 435-457.
-
16R.S.O. Nunes & W.D. Bastos. Energy decay for the linear Klein-Gordon equation and boundary control. J. Math. Anal. Appl. , 414 (2014), 934-944.
-
17R.S.O. Nunes & W.D. Bastos. Analyticity and near optimal time boundary controllability for the linear Klein-Gordon equation. J. Math. Anal. Appl. , 445 (2017), 394-406.
-
18D.L. Russell. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math., 52 (1973), 189-211.
-
19D. Tataru. On regularity of the boundary traces for the wave. Ann. Scuola Norm. Pisa, C. L. Sci., 26 (1998), 185-206.
Publication Dates
-
Publication in this collection
03 Aug 2020 -
Date of issue
May-Aug 2020
History
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Received
18 June 2019 -
Accepted
30 Apr 2020