Analysis and explicit solvability of degenerate tensorial problems
 Tongxing Li^{1} and
 Giuseppe Viglialoro^{2}Email authorView ORCID ID profile
Received: 6 November 2017
Accepted: 15 December 2017
Published: 4 January 2018
Abstract
We study a twodimensional boundary value problem described by a tensorial equation in a bounded domain. Once its more general definition is given, we conclude that its analysis is linked to the resolution of an overdetermined hyperbolic problem and hence present some discussions and considerations. Secondly, for a simplified version of the original formulation, which leads to a degenerate problem on a rectangle, we prove the existence and uniqueness of a solution under proper assumptions on the data.
Keywords
MSC
1 Introduction, motivation, and structure of the paper
It is well known that the Dirichlet problem associated to a hyperbolic equation is often employed as an example of an ill posed problem in the theory of hyperbolic partial differential equations (see [1]). Nevertheless, a wide range of real problems arising in nature (gas dynamics, torsion theory of shells with alternating sign curvature, mechanical behaviors of bending structures, etc.) are in fact mathematically described through hyperbolic equations; thereafter, a deserving undertaking is developing a casuistry for which such problems are, indeed, well posed.
 (H1):

the graph of z represents an almost everywhere (a.e.) negative Gaussian curvature (alternating sign curvature) surface,^{2} that is,$$ z_{,xx}z_{,yy}z_{,xy}^{2} < 0 \quad\text{a.e. in } \bar{\Omega}; $$
 (H2):

the tensor σ is almost everywhere positive definite,^{3} that is,$$ \sigma_{xx}\xi_{1}^{2}+2\sigma_{xy} \xi_{1} \xi_{2}+\sigma_{yy}\xi_{2}^{2}> 0 \quad\forall (\xi_{1},\xi_{2}) \neq(0,0) \text{ and a.e. in } \bar {\Omega}. $$
 \((\mathcal{HP})\) :

a problem of hyperbolic type where the tensor σ is the unknown: given a function z with a.e. negative Gaussian curvature, find an a.e. positive definite tensor σ fulfilling (1);
The remaining structure of the paper is drawn as follows. In Section 2, we formulate the socalled General Problem associated with (\(\mathcal{HP}\)), which is a very broad (tensorial) boundary value problem modeling an optimal mechanical scenario appearing in membrane structures. As detailed in [4, 5, 7], the boundary of the domain is split into two parts; on a portion, mechanically corresponding to the boundary of the membrane tensioned by rigid elements (which admit any geometrical shape), a Dirichlet boundary condition is assumed, whereas on the remaining part, associated with the complementary boundary of the membrane tensioned by cables (which can be neither straight lines nor changing curvature curves), an unusual boundary relation is given. We discuss the main mathematical properties of this formulation, also in terms of other wellknown results, and we conclude that this is an overdetermined, generally illposed, problem, for which the part of the domain with the singular boundary condition (free boundary) plays the role of a further unknown. In addition, Section 3 deals with the analytical resolution of the Reduced Problem, a simplified version of the General Problem, linked to a more restrictive physical situation, where the membrane is only tensioned by rigid elements: we examine a specific case in a rectangle, for which the resulting Dirichlet boundary problem admits an explicit unique solution. Specifically, once a polynomial for the function z is fixed in such a way that its graph identifies a surface with a.e. negative Gaussian curvature, by manipulating the tensorial expressions of the problem the main equation reads \(cy^{2(n1)}\sigma _{yy,xx}\sigma_{yy,yy}=0\) in \((0,a)\times(b,0)\) with some \(a,b,c>0\) and n an integer greater than 1, exactly degenerating for \(y=0\). Connected to the last partial differential equation (PDE), the question of well posedness of boundary value problems for linear secondorder PDEs of the form \(\psi(y)u_{,xx}u_{,yy}=0\), where ψ is a sufficiently regular function with specific properties, has been studied in several works: contributions as [8–11] (and references therein) include discussions concerning the notorious special case of the mixed elliptichyperbolic Tricomi equation, obtained for \(\psi(y)=y\), and provide a general comprehensive picture of the whole analysis. Also in line with these works, we cite paper [12], employed in this present investigation to prove the main result asserted in Theorem 3.1 of Section 3.2 and, in particular, to construct the claimed explicit solution σ to system (1). Finally, to mathematically point out the different physical behaviors between shells and membranes, in Section 3.3, we also solve the same Reduced Problem presented in Section 3.1 but in the case where no restriction on the sign of σ is required (Theorem 3.2); besides (in Section 3.4), we give a graphical representation of the derived solutions corresponding to the two mechanical situations. The closing Section 4 provides some final considerations.
2 The general problem
The following section includes some necessary tools used to our main purposes.
2.1 Definition of the domain and the boundary data
Assumptions 2.1
 (i)\(\Gamma^{c}\) is represented by a regular curve in \(\mathbb{ R}^{2}\) with no vanishing curvature whose parameterization is given by \(\gamma (t)=(x(t),y(t))\), \(t\in[t_{0},t_{1}]\), and obtained by solving the ordinary differential equation$$ z_{,xx}\bigl(x'\bigr)^{2}+2z_{,xy}x'y'+z_{,yy} \bigl(y'\bigr)^{2}=0. $$(2)
 (ii)
\(\Gamma^{r}\) is arbitrarily fixed but such that \(\Gamma^{r} \cap \Gamma^{c}=\{P_{0},P_{1}\}\), where \(P_{0}=\gamma(t_{0})\) and \(P_{1}=\gamma(t_{1})\).
 (iii)
n is the outward unit vector to Γ.
 (iv)
\(\mathbf{f}^{r}=(f^{r}_{1},f^{r}_{2})\) and \(\mathbf{f}^{c}=(f^{c}_{1},f^{c}_{2})\) are two regular vectorial fields, per unit length, defined on \(\Gamma^{r}\) and \(\Gamma^{c}\), respectively; in addition, the continuity conditions \(\mathbf{f}^{r}(P_{0})=\mathbf{f}^{c}(P_{0}) \) and \(\mathbf{f}^{r}(P_{1})=\mathbf{f}^{c}(P_{1}) \) have to be satisfied.
2.2 Mathematical formulation of the general problem
Let us now describe the details of the General Problem we are interested in.
General Problem 1
2.3 Analysis and discussion of the general problem
Since the vectorial field \(\mathbf{f}^{c}\) has to satisfy both expressions (3e) and (4), it cannot be uniquely and arbitrarily assigned (see Counterexample 1). Essentially, this singularity is tied to the fact that the unknowns σ and g are coupled through \(\mathbf{f}^{c}\) and that, even more, they are especially linked to the domain \(\Gamma^{c}\); subsequently, the General Problem 1 represents an overdetermined system that commonly admits no solutions.
Moreover, so far we did not manage to derive a nontrivial analytical solution to the same problem; indeed, the question how to fix \((z,\boldsymbol {\sigma},\gamma, \mathbf{f}^{c},\mathbf{f}^{r},g,g_{0})\) such that all relations (2), (1), and (4) hold seems rather challenging. In line with this, the case where z is a linear function has no mathematical interest (and even less physical), since relation (3c) is automatically satisfied and, in addition, any \(\gamma(t)=(x(t),y(t))\) is compatible with condition (2); hence the problem merely loses its intrinsic nature. The same mathematical and physical reasons make that functions behaving as \(z(x,y)=\alpha^{2} x^{2}\beta^{2} y^{2}\) with \(\alpha ,\beta\in \mathbb{ R}_{0}\) do not lead to stimulating issues, since (2) would infer straight lines parameterized as \(\gamma(t)= (t,\mp(\alpha/\beta)t+\text{constant} )\) for \(\Gamma^{c}\), and essentially the General Problem would ‘degenerate’ to the forthcoming Reduced Problem (see p. 6).
Returning to overdetermined boundary value problems, there exists a large amount of literature dealing with the subject; in general, these problems are prescribed by a classical partial differential equation where both Dirichlet and Neumann boundary conditions are imposed on the boundary of the domain. Some meriting questions about the analysis are the proof of the existence of solutions, possibly uniqueness, and the study of their properties. The main characteristic of the overdetermined problems is that such an overdetermination makes the domain itself unknown (free boundary problems), or in general it cannot be arbitrarily assigned, resulting solvability only in precise domains; beyond the landmark result by Serrin [13], we refer also to [14–17] for contributions regarding both elliptic and hyperbolic equations.
Remark 1
As to the specific problem we are focusing on here, let us quote that the elliptic version of the General Problem 1, herein indicated with (\(\mathcal{EP}\)) and briefly defined in Section 1, represents the wellknown formfinding problem of tensegrity structures. In line with this, there is a large literature concerning the analysis and the design of selfsupporting membrane structures, and they are essentially based on appropriate discrete numerical methods (see, for instance, [18–20] and references therein).
On the other hand, as far as the continuous approach of problem (\(\mathcal{EP}\)) is concerned, this has been deeply discussed by one of the authors of this paper in recent investigations. We mention that the complete formulation of problem (\(\mathcal{EP}\)) corresponds to a boundary value problem, in the unknown z, described by an elliptic differential equation in Ω. The portion \(\Gamma ^{c}\) of Γ is indeed constructed by means of σ (which in this case is fixed) and \(\mathbf{f}^{c}\). Finally, the whole Γ is endowed with Dirichlet boundary conditions, but, in accordance to overdetermined problems, on \(\Gamma^{c}\) another relation involving \(z_{,y}\) and replacing expression (2) has to be satisfied as well. The technical aspects for the construction of \(\Gamma^{c}\) and the definition of the complete boundary value problem are available in [21] and [7]; in particular, as for the General Problem, the questions of the existence and derivation of an explicit solution are still open. Conversely, in the last two aforementioned contributions an equivalent number of numerical procedures exactly tied to free boundary approaches are proposed and employed as resolution methods.
Counter example 1
(Ill posedness of the General Problem)
Let us fix \(z(x,y)=A^{2}x^{4}+6B^{2}y^{2}\) (with \(A,B\in \mathbb{ R}_{0}\)). By equation (2) we can choose as \(\Gamma^{c}\) the curve \(\gamma (t)=(t,At^{2}/(2B))\), \(t\in[t_{0},t_{1}]\). In addition, \(\sigma_{xx}=1\), \(\sigma_{xy}=\sigma_{yx}=0\), and \(\sigma_{yy}=A^{2}x^{2}/B^{2}\) is a symmetric and positive definite tensor a.e. in \(\mathbb{ R}^{2}\) that solves equations (3a), (3b), and (3c). As to the expression of \(\mathbf{f}^{c}\), since \(\mathbf{n}=(At/B,1)/ \Vert \gamma'(t) \Vert \), relation (3e) infers \(\mathbf{f}^{c}=(At/B,A^{2}t^{2}/B^{2})\) on \(\Gamma^{c}\); thereafter from the first and last conditions of (4) we arrive at \(g(t)=At^{2}/(2B)+g_{0}+At_{0}^{2}/(2B)\), which, in view of the second relation of (4), leads to the incongruence \(3A^{2}t^{2}/(2B^{2})+g_{0}A/B+A^{2}t_{0}^{2}/(2B^{2})=A^{2}t^{2}/B^{2}\) for all \(t\in[t_{0},t_{1}]\).
3 The reduced problem
3.1 Mathematical formulation of the reduced problem
Let us now introduce the Reduced Problem; essentially, its definition corresponds to setting \(\Gamma^{c}=\emptyset\) in Assumptions 2.1. Therefore, \(\Gamma^{c}\), \(\mathbf{f}^{c}\), and g do not take part in the formulation, and, subsequently, we have \(\Gamma =\Gamma^{r}=\partial\Omega\); moreover, for convenience, we avoid the superscript r for \(\mathbf{f}^{r}\), and we directly consider f as a given vectorial field, per unit length, on \(\Gamma=\partial \Omega\).
Reduced Problem 1
In the rest of this section, we show the existence and uniqueness of a solution to the Reduced Problem 1 defined in a rectangle.
3.2 A case of explicit resolution in a rectangle
Theorem 3.1
3.3 The case of no restriction on the sign definiteness of σ
By retracing the proof of Theorem 3.1 we observe that, behind other technical reasons, the final expression of the solution σ derived in (18) is deeply tied to the requirement of the a.e. positivity definiteness of such a tensor; conversely, as announced in the introductory comments of Section 1, if this restriction is omitted, then for the same function \(z(x,y)=c_{1}x^{2}c_{2}y^{2n}\), the unique solution in Ω̄ exhibits a more general representation, precisely given by (16) and (17). Subsequently, we have this further result, which we state without further comments.
Theorem 3.2
3.4 Two specific examples: representation of the solution
More precisely, for \(K=5\) in expression (18), Figure 2 represents the case of the equilibrium between the stress and shape of a membrane structure. We can realize that the component \(\sigma_{xx}\) is positive a.e. in Ω and increases as \(y\rightarrowb\) and constant values of x (see the below part of the topright corner of Figure 2); in the limit, it exactly corresponds to a zone on the membrane with major tension, along the xdirection, with respect to others (Figure 2, above part, topright). As to \(\sigma_{yy}\), it is constant and positive in Ω, so that the corresponding tension along the ydirection is uniformly distributed on the surface (see the lowerleft corner of Figure 2); finally, the lowerright corner of Figure 2 highlights the nil contribution of \(\sigma _{yx}=0\) in Ω, that is, the absence of shear stress on the membrane.
Conversely, if in (19) we set \(\lambda =4\), \(K=0.5\), and \(H=2\), the features of the solution σ are summarized in Figure 3, which models the balance between the stress and shape for a shell structure. Relaxing the assumption on the sign definiteness of σ, we obtain not only positive expressions for the components \(\sigma_{xx}\) and \(\sigma_{yy}\) on the whole Ω, but also regions of the rectangle where they are negative (see the below part of the topright and lowerleft corner of Figure 3, respectively); this aspect identifies zones of the shell where tensions or compressions are present along both the x and ydirections (same corners of Figure 3, but the above part).
By the above we stress again that the general solution for the tensor σ given by relation (18) represents a very particular and simplified case of solution (19). Such a leap has not to appear surprising since, indeed, it is intimately linked to the different natures of the problems: in particular, when a membrane is considered, a strong limitation on the state of its stress tensor that exactly balances its shape is naturally expected and absolutely consistent with the mechanical problem.
4 Conclusions
This paper is devoted to a twodimensional boundary value system described by a tensorial equation in a bounded domain. Its more general definition leads to the resolution of an overdetermined hyperbolic problem, whose analysis is complex and represents a challenging open question in the field. Indeed, for a simplified version, whose formulation is given by a degenerate problem on a rectangle, the existence and uniqueness of a solution under proper assumptions on the data can be proven. Behind its pure mathematical interest, this research is motivated by natural applications to real mechanic problems, linked to the equilibrium of membrane and shell structures. In this sense, the derived solutions achieved throughout the paper are totally consistent with the expected results.
In this paper the partial derivative of a function f with respect to a certain variable w is indicated with \(f_{,w}\); similar symbols concerning higherorder derivatives (double or mixed) are introduced in a natural way.
With some abuse of language, we also use sentences as z has an a.e. negative Gaussian curvature or z is a function with a.e. negative Gaussian curvature or similar; in any case, no misunderstanding will be possible from the context.
Obviously, if \(\sigma_{xx}\xi_{1}^{2}+2\sigma_{xy}\xi_{1} \xi _{2}+\sigma_{yy}\xi_{2}^{2}< 0\ \forall (\xi_{1},\xi_{2}) \neq(0,0)\) and a.e. in Ω̄, we say that the tensor σ is almost everywhere negative definite. We say that the tensor is indefinite (or alternating sign definite) a.e. in Ω̄ if it is a.e. negative definite for some values of Ω̄ and a.e. positive definite for the others.
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors and anonymous referee for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. Giuseppe Viglialoro is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Funding
The research of Tongxing Li is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China. Giuseppe Viglialoro acknowledges the Italian Ministry of Education, University and Research (MIUR) for the financial support of Scientific Project ‘Smart Cities and Communities and Social Innovation  ILEARNTV anywhere, anytime  SCN_00307’.
Authors’ contributions
Both authors contributed equally to this work. They both read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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References
 Hörmander, L: Linear Partial Differential Operators. Springer, Berlin (1963) View ArticleMATHGoogle Scholar
 Timoshenko, S, WoinowskyKrieger, S: Theory of Plates and Shells. McGrawHill, New York (1959) MATHGoogle Scholar
 Ventsel, E, Krauthammer, T: Thin Plates and Shells. Theory, Analysis, and Applications. Marcel Dekker, New York (2001) View ArticleGoogle Scholar
 Viglialoro, G, Murcia, J, Martínez, F: Equilibrium problems in membrane structures with rigid boundaries. Inf. Constr. 61(516), 5766 (2009) View ArticleGoogle Scholar
 Viglialoro, G, Murcia, J: Equilibrium problems in membrane structures with rigid and cable boundaries. Inf. Constr. 63(524), 4957 (2011) View ArticleGoogle Scholar
 Viglialoro, G, Murcia, J, Martínez, F: The 2D continuous analysis versus the density force method (discrete) for structural membrane equilibrium. Inf. Constr. 65(531), 349358 (2013) View ArticleGoogle Scholar
 Viglialoro, G, González, Á, Murcia, J: A mixed finiteelement finitedifference method to solve the equilibrium equations of a prestressed membrane having boundary cables. Int. J. Comput. Math. 94(5), 933945 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Agmon, S, Nirenberg, L, Protter, MH: A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptichyperbolic type. Commun. Pure Appl. Math. 6, 455470 (1953) MathSciNetView ArticleMATHGoogle Scholar
 Hačev, MM: The Dirichlet problem for the Tricomi equation in a rectangle. Differ. Uravn. 11, 151160 (1975) 205 (in Russian) MathSciNetMATHGoogle Scholar
 Tricomi, F: Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 5(14), 134247 (1923) MATHGoogle Scholar
 Lupo, D, Morawets, CS, Payne, KR: On closed boundary value problems for equations of mixed elliptichyperbolic type. Commun. Pure Appl. Math. 60(9), 13191348 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Khachev, MM: The Dirichlet problem for a degenerate hyperbolic equation in a rectangle. Differ. Equ. 37(4), 603606 (2001) MathSciNetView ArticleMATHGoogle Scholar
 Serrin, J: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43(4), 304318 (1971) MathSciNetView ArticleMATHGoogle Scholar
 Agostiniani, V, Magnanini, R: Symmetries in an overdetermined problem for the Green’s function. Discrete Contin. Dyn. Syst., Ser. S 4(4), 791800 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Henrot, A, Philippin, GA: Some overdetermined boundary value problems with elliptical free boundaries. SIAM J. Math. Anal. 29(2), 309320 (1998) MathSciNetView ArticleMATHGoogle Scholar
 Nacinovich, M: Overdetermined hyperbolic systems on l.e. convex sets. Rend. Semin. Mat. Univ. Padova 83, 107132 (1990) MathSciNetMATHGoogle Scholar
 John, F: The Dirichlet problem for a hyperbolic equation. Am. J. Math. 63(1), 141154 (1941) MathSciNetView ArticleMATHGoogle Scholar
 Argyris, JH, Angelopoulos, T, Bichat, B: A general method for the shape finding of lightweight tension structures. Comput. Methods Appl. Mech. Eng. 3(1), 135149 (1974) View ArticleGoogle Scholar
 Linkwitz, K: About formfinding of doublecurved structures. Eng. Struct. 21(8), 709718 (1999) View ArticleGoogle Scholar
 Veenendaal, D, Block, P: An overview and comparison of structural form finding methods for general networks. Int. J. Solids Struct. 49(26), 37413753 (2012) View ArticleGoogle Scholar
 Viglialoro, G, Murcia, J: A singular elliptic problem related to the membrane equilibrium equations. Int. J. Comput. Math. 90(10), 21852196 (2013) View ArticleMATHGoogle Scholar