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I want to show the existence of solutions to an overdetermined system of second-order partial differential inequalities in a given region $\Omega\subset\mathbb{R}^2$, \begin{equation*} \begin{cases} P_1(u)\geq 0,&\\ P_2(u)\geq 0,&\\ P_3(u)\geq 0.& \end{cases} \end{equation*}

Updated: $P_i$ are linear/quasilinear/nonlinear second-order differential operators, \begin{equation*} \begin{cases} P_1(u)=&\hspace{-0.3cm}xu_{xx}+yu_{xy}+x^2u_{yy}+4u_x.\\ P_2(u)=&\hspace{-0.3cm}u_{xx}u_{yy}-u_{xy}^2+(xu_x-yu_y)u_{xx}+(yu_x-x^2u_y)u_{xy}+x(xu_x-yu_y)u_{yy}.\\ P_3(u)=&\hspace{-0.3cm}(u_{xx}u_{yy}-u_{xy}^2)u_x+(xu_x^2-yu_xu_y-x^2u_y^2)u_{xx}+(yu_x^2-x^2u_xu_y-xyu_y^2)u_{xy}+(x^2u_x^2-xyu_xu_y-y^2u_y^2)u_{yy}. \end{cases} \end{equation*}

To do this, I am trying to solve for $(u,f_1,f_2,f_3)$ of the following system of PDEs, \begin{equation*} \begin{cases} P_1(u)=f_1^2,&\\ P_2(u)=f_2^2,&\\ P_3(u)=f_3^2.& \end{cases} \end{equation*}

Could anyone suggest any references to deal with either system? Any suggestion is appreciated.

Updated: Is it true that if I find all compatible conditions for $f_1,f_2,f_3$ and they are satisfied, then this system has a solution?

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  • $\begingroup$ It's important here to say what space of functions you are trying to solve these inequalities in. It looks like $C^2(\Omega)$ might work. These operators appear to be variants of the Monge-Ampère operator. In that case, you might want to start by checking if convex functions or some variant of them satisfy these inequalities. $\endgroup$ – Deane Yang May 20 '17 at 16:39
  • $\begingroup$ Because your equations don't have any explicit $u$-dependence, you are really asking about inequalities for Lagrangian surfaces in $xypq$-space (where, in the traditional notation, $p=u_x$ and $q=u_y$). However, you must be looking for functions $u$ that satisfy some additional conditions, since $u(x,y)\equiv0$ obviously solves the inequalities in any region $\Omega$. In fact, $u(x,y) = f(x)$ where $f'(x)>0$ and $xf''(x)\ge0$ also solves the inequalities. You should state what your other conditions are. It would also help to motivate this question, which, as written, seems awfully random. $\endgroup$ – Robert Bryant May 26 '17 at 14:01
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Existence theorems for real analytic pde systems were developed by Riquier and Janet in the first quarter of the twentieth century (see, for example, Riquier's 1910 book Les systèmes d'équations aux dérivées partielles).

An up to date treatment of their work, its modernisation by Pommaret, and its subsequent extensions is given in Seiler's book Involution The formal theory of differential equations and its applications in computer algebra. A lot of progress in 100 years.

Edit: an application of these ideas to overdetermined systems in particular can be found in Kunio Kakié "On Regularity of Solutions to Overdetermined Non Linear Partial Differential Equations", Commetarii Mathematici Universitatis Sancti Pauli Vol. 52, No. 2 2003, pp. 125-138. Note that as per the comments added below, this paper discusses local properties only.

Edit(2): regarding your question about compatibility conditions for the $f$ functions, the answer is "highly likely, but not guaranteed". You need your system to be involutive. If it isn't, then the system has to prolonged (differentiated to produce a higher order system); the Cartan-Kuranishi theorem states that a finite number of prolongations will produce either an involutive system, or an inconsistency. This is explained in Seiler's book mentioned previously, which also includes worked examples.

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  • $\begingroup$ Unfortunately, the results are only local. $\endgroup$ – Ben McKay May 19 '17 at 13:55
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Because some of your operators are not linear, I don't have a definitive answer. But Haim Brézis considered problems of the form $$\max(A_1u-f_1,A_2u-f_2)=0$$ where $A_1,A_2$ are second-order elliptic operators. See his paper The Hamilton-Jacobi-Bellman equations and variational inequalities. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 385–395, Pitagora, Bologna, 1979. See also

(Brézis + L. C. Evans) in Arch. Rat. Mech. Anal. 70 (1979), no. 1, 1–13,

P.-L. Lions. Some problems related to the Bellman-Dirichlet equation for two operators. Comm. Partial Differential Equations 5 (1980), no. 7, 753–771.

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The Cartan--Kaehler theorem gives sufficient conditions for the existence of local solutions of any real analytic system of partial differential equations. However, the solutions are only local. See Bryant, Chern, Gardiner, Goldschmidt and Griffiths, Exterior Differential Systems, MSRI.

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  • $\begingroup$ Thank you very much. Is there any method to show the existence of solutions to a PDE system in some region? $\endgroup$ – Denifer May 19 '17 at 17:20
  • $\begingroup$ Does that mean that currently there is no method to deal with this knid of problem? $\endgroup$ – Denifer May 19 '17 at 23:12
  • $\begingroup$ I don't know of a method to solve the problem directly. You might want to start by checking local existence, since that is not obvious. You might then, if you are lucky and clever, find that some argument like the method of characteristics, i.e. the characteristic variety of your equation system may give a family of curves on solutions. Identify how to construct those curves, just as a family of curves in the plane, independently of any particular choice of solution. Then guess how to construct all solutions out of those curves. But that is all speculation, and might come to nothing. $\endgroup$ – Ben McKay May 20 '17 at 16:42

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