COMPARISON THEOREMS FOR ULTRAHYPERBOLIC EQUATIONS

Sturmian comparison theorems are obtained for solutions of a class of normal and singular ultrahyperbolic partial differential equations. In the singular case, solutions are considered which satisfy non-standard boundary conditions.

I. Introduction.The Sturmian comparison theorems for ordinary differential equations have been extended extensively to partial differ- ential equations of the elliptic type.For example, see Kuks [13,Swanson [2], [3], Diaz and McLaughlin [4], and Kreith and Travis [5], to mention only a few.By employing Swanson's technique, Dunninger [63 obtained a comparison theorem for parabolic partial differential equations.
His result was recently generalized by Chan and Young [7] to time-dependent quasilinear differential systems.However, for partial differential equations of the hyperbolic type, very little is known.In fact, as far as these authors know, the only comparison results known for hyperbolic equations were obtained just recently by Kreith  [8], Travis [9], [i0], and Young [ii].
In this paper, we shall present some comparison theorems for the pair of normal ultrahyperbolic equations (Aij (X)Ux)x.(aij (y)u + p(y)u 0 i 3 Yi Yj (Bij(X)Vx.)x.(bij(y)Vy) + q(y)v 0 1 J i Yj and the pair of singular ultrahyperbolic equations x (aij , where H and G are bounded regular domains in E n and Em, respectively.For brevity, we let x (x I ,x n) denote a point in H and Y (Yl ,ym a point in G.Moreover, we adopt Einstein's summation convention concerning repeated indices. The matrices (Aij), (aij), (Bij) and (bij) are all assumed to be symmetric and positive definite with continuously differentiable elements in their respective domains of definition.The coefficients p and q are continuous in G, while the a.'s are real parameters, 1 i= l,...,n.
We associate with the above equations the following eigenvalue problems (i.I) and -(aijyi )yj + pC X in G + r(y) 0 on G (bijyi)y j + q in G -+ s(y) 0 on G 3n b where r and s are continuous functions on G, and --a j, --b Cyi na ij Yi n b ij j (Vl 'Vm) being the outward unit normal vector on 3G.
2. The Normal Case.We consider the following boundary value problems ( where at least one strict inequality holds, then every solution v e C2{D) of {2.2) has a zero in D.
Proof.Suppose v is a solution of {2.2).Let 0 and 0 be the positive eigenfunctions [12] corresponding to the first eigenvalues X 0 and 0 of (1.1) and {1.2), respectively.Define U(x) G Since u and v satisfy (2.1) and (2.2), respectively, and 0 and 0 are eigenfunctions of (1.1) and (1.2), respectively, it follows by the divergence theorem that (2.4) (BijVx.) 1 By the variational characterization of the eigenvalues of (i.i) and (1.2), it follows from the assumptions (2.3) that 0 < 0" Hence, by Theorem i0 of [4], V has a zero in H, say at x x 0.Then, since 40 > 0 in G, the equation is a zero of v in D, and the theorem is proved.
We remark that if the solution u of (2.1) is required to satisfy the boundary condition A..u t. + R(x)u 0 ij x i j on H x G, instead of u 0, the conclusion of Theorem 2.1 remains valid for solutions v of (2.2) satisfying the additional condition B..v t. + S(x)v 0 i x i on H x G, where R(x) <_ S(x) and (tl,...,t) is the outward unit n normal vector on H.This is a consequence of Theorem 2 of [3] where now H is the domain l,...,n, and assume that 1 (3.3) (aij) < (bij), p < q, r < s in G. G n where as usual dx dx l...dxn and dy dy l...dy m.
Proof.As in the proof of Theorem 2.1, we let 0 and 0 be the positive, normalized, eigenfunctions of (1.1) and (1.2) corresponding to the first eigenvalues XO and UO" Then the functions U and V, defined as in the proof of Theorem 2.1, satisfy the equations   X 0 +...+ X and U0 i +''" + Un' where X1 n the roots of the equation the X.'s being J(l_ai)/2( Xiai) 0, (i l,...,n) 's are arbitrary constants.Here J (t) denotes the Bessel and the i p function of the first kind of order p.
From condition (3.3), it follows that X0 < u0" Hence there exists an integer j such that X. < u..This implies that V vanishes along the line x./X-./.a. < a.. Since 0(y) > 0 in G we conclude by 3 3 3 3 the same argument as in the proof of Theorem 2.1 that V has a zero in D. The proof of this theorem is similar to that of Theorem 3.1 and is therefore omitted.
We conclude this paper with a theorem that is valid for all values of the parameters ..

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: .

3 .
The Singular Case.We consider the singular boundary value is a solution of (3.1) which is positive in D and satisfies the conditions (3.4) fa I x=[u(x'y) ) / 2 (.]. x

Theorem 3 . 2 .
Let a. >-i, i 1 n, and assume that condition (3.3) holds.If u is a solution of (3.1) which is positive in D

Theorem 3 .
3. Let < .< , i 1 n, and assume that (3.3) i holds.If u is a solution of (3.1) which is positive in D and satisfies then every solution v of (3.2) satisfying ]v(O,y)[ < , y has a zero in D.