Complex Variable And Elliptic Equations

[Suyay Escarsega]

Wellreal-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.

As Qiaochu says, one answer is elliptic regularity: complex differentiable functions obey a non-trivial equation (the Cauchy-Riemann equation) which implies a integral representation (the Cauchy integral formula) which then implies analyticity (Taylor expansion of the Cauchy kernel); the ellipticity of the Cauchy-Riemann equation is what gives the analyticity of its fundamental solution, the Cauchy kernel. Real differentiable functions obey no such equation.

Another approach is via Cauchy's theorem. In both the real and complex setting, differentiability implies that the integral over a closed (or more precisely, exact) contour is zero. But in the real case this conclusion has trivial content because all closed contours are degenerate in one (topological) dimension. In the complex case we have non-trivial closed contours, and this makes all the difference.

EDIT: Actually, the above two answers are basically equivalent; the latter is basically the integral form of the former (Morera's theorem). Also, to be truly nitpicky, "differentiable" should be "continuously differentiable" in the above discussion.

One answer is that property of being complex analytic is equivalent to being a solution to a differential equation (namely the Cauchy-Riemann equations) whereas there is no analagous formulation for being a smooth function. Once you have this formulation it should be immediately clear that complex analytic functions are rigid because solutions to differential equations are rigid. The whole idea of boundary value problems and initial value problems is predicated on the fact that knowing a solution to differential equation in a small area determines its values everywhere, which is precisely the type of rigidity that complex analytic functions have.

So you might ask why do complex analytic functions satisfy the Cauchy-Riemann equations. Well every real smooth function from R^2 to R^2 has a derivative, which is a 2x2 matrix. Requiring the function to be complex differentiable is the same as requiring that matrix to be a "complex number", i.e. a matrix of the following form: first row = [a, -b], second row = [b, a]. Well this condition is precisely the Cauchy-Riemann equations.

The book Visual Complex Analysis gives a good explanation: locally, analytic functions are rotations and dilations. Disks go to disks. A smooth function of two real variables may map disks to ellipses. That is, a real valued function can distort disks in a way that analytic functions cannot.

To expand on Dinakar's comment about boundary value problems, the physical intuition one should have here is that the real and complex parts of a complex differentiable functions are harmonic functions (this is just a restatement of the C-R equations). An important way harmonic functions arise is as solutions of the steady-state heat equation, so one can think of the values of a harmonic function on a contour as temperatures and the values of a harmonic function in the interior of the contour as the steady-state distribution of temperature determined by the distribution of temperature on the contour. When the contour is a circle one can compute this distribution by convolving with the Poisson kernel; this is a special case of Cauchy's integral formula. The Poisson kernel itself is the canonical example of a "good kernel," and it finds application in Fourier analysis for that reason. One generally expects convolution by a good kernel to have nice properties.

In turn, one reason why diffusion should have anything to do with complex differentiability is that in both cases one wants path integrals to be homotopy invariant, in the first case because path integrals should give the difference in potential and in the second case because this is the correct generalization of the fundamental theorem of calculus.

In the real case, h is real, and approaches 0 along the real line, from the right and from the left. In the complex case, h is complex, and approaches 0 from any possible direction. This makes it more difficult for the limit to exist, and thus for complex functions of a complex variable to have a derivative. When such a function is viewed as a pair of real functions of two real variables, that is, when we write

A complex function is analytic if and only if locally it can be represented by a power series. This means that (at least locally) an analytic function is determined by countable data (namely, the Taylor coefficients of its local expansion). This is not true for smooth functions on the real line. In my thinking the rigidity of analytic functions stems from this countable determinacy.

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn \displaystyle \operatorname sn for sin \displaystyle \sin . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular,[1] but his work was published much later.

There is a definition, relating the elliptic functions to the inverse of the incomplete elliptic integral of the first kind F \displaystyle F . These functions take the parameters u \displaystyle u and m \displaystyle m as inputs. The φ \displaystyle \varphi that satisfies

(the incomplete elliptic integral of the first kind) is computed. On the unit circle ( a = b = 1 \displaystyle a=b=1 ), u \displaystyle u would be an arc length.The quantity u [ φ , k ] = u ( φ , k 2 ) \displaystyle u[\varphi ,k]=u(\varphi ,k^2) is related to the incomplete elliptic integral of the second kind (with modulus k \displaystyle k ) by[8]

is the transformation of m in the imaginary transformation, then the other transformations can be built up by successive application of these two basic transformations, yielding only three more possibilities:

where i = U, I, IR, R, RI, or RIR, identifying the transformation, γi is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other nine transformed functions can be built up from the above three. The reason the cs, ns, ds functions were chosen to represent the transformation is that the other functions will be ratios of these three (except for their inverses) and the multiplication factors will cancel.

From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions[3]

In conjunction with the addition theorems for elliptic functions (which hold for complex numbers in general) and the Jacobi transformations, the method of computation described above can be used to compute all Jacobi elliptic functions in the whole complex plane.

The regularity of the solution of this equation is studied in the literature by many authors and by different tools. It was first proved in pioneering work [4] that the solution u to the Dirichlet problem in a ball B belongs to \(C^1,1(B) \cap C^0(\overlineB)\) provided that the right-hand side \(f \in C^2(\overlineB)\) and the boundary data is \(C^2(\partial B)\). Another important regularity result concerning the Dirichlet problem associated with (2) in a strictly pseudoconvex domain, established in [5], is the smooth regularity up to the boundary of its solution when the right-hand side, the boundary data and the domain are all smooth (the strict positivity of the right-hand side is also essential).

The local regularity of (2) (no boundary data) was also studied. A sharp result was obtained in [2] by developing some methods in [24]: solutions \(u \in W^2,p_\mathrm loc\) of (2) with \(f \in C^\infty \) are necessarily \(C^\infty \) whenever \(p>n(n-1)\), and no smaller exponent p can be expected in general.

The local regularity of other equations of the form (1) were also studied. Notably, in [9], a counterpart of [2] was proved for the so-called complex k-Hessian equation under the assumption that u belongs to \(W^2,p_\mathrm loc\) with \(p> n(k-1)\).

The goal of the present paper is to extend the approach of [2] and [9] to more general nonlinear complex degenerate elliptic equations. We will introduce simple conditions on the nonlinearity F to obtain general local regularity results, thus considerably broadening the field of application of this method.

Interior estimates for this equation in the real setting have been studied recently in [7]. In the complex setting this operator was discussed in [23] and, in the special case \(k=n-1\), also in [25]. It has been shown in [8] that this operator does not satisfy an integral comparison principle, which makes the associated potential theory much harder to be developed. Finally, let us also mention that the Dirichlet problem associated with this operator was studied in [28] using a probabilistic approach.

The outline of the paper is as follows. After recalling some basic notations, we precise in Sect. 1.2 what kind of nonlinear operators F are considered in this article and we present our main results. Their proofs are the purpose of Sect. 2. Some examples of Hessian equations covered by such a framework are then given in Sect. 3. Finally, in Sect. 4 we detail the case study of the \(\mathrm MA_k\)-equation.

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