In
mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
and uniqueness theorem for
analytic partial differential equations associated with
Cauchy initial value problem In mathematics, a Cauchy () boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A C ...
s. A special case was proven by , and the full result by .
First order Cauchy–Kovalevskaya theorem
This theorem is about the existence of solutions to a system of ''m'' differential equations in ''n'' dimensions when the coefficients are
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s. The theorem and its proof are valid for analytic functions of either real or complex variables.
Let ''K'' denote either the
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
of real or complex numbers, and let ''V'' = ''K''
''m'' and ''W'' = ''K''
''n''. Let ''A''
1, ..., ''A''
''n''−1 be
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s defined on some
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of (0, 0) in ''W'' × ''V'' and taking values in the ''m'' × ''m'' matrices, and let ''b'' be an analytic function with values in ''V'' defined on the same neighbourhood. Then there is a neighbourhood of 0 in ''W'' on which the
quasilinear
Quasilinear may refer to:
* Quasilinear function, a function that is both quasiconvex and quasiconcave
* Quasilinear utility, an economic utility function linear in one argument
* In complexity theory and mathematics, O(''n'' log ''n'') or some ...
Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value prob ...
:
with initial condition
:
on the hypersurface
:
has a unique analytic solution ''ƒ'' : ''W'' → ''V'' near 0.
Lewy's example shows that the theorem is not more generally valid for all smooth functions.
The theorem can also be stated in abstract (real or complex) vector spaces. Let ''V'' and ''W'' be finite-dimensional real or complex vector spaces, with ''n'' = dim ''W''. Let ''A''
1, ..., ''A''
''n''−1 be
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s with values in
End (''V'') and ''b'' an analytic function with values in ''V'', defined on some
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of (0, 0) in ''W'' × ''V''. In this case, the same result holds.
Proof by analytic majorization
Both sides of the
partial differential equation can be expanded as
formal power series and give recurrence relations for the coefficients of the formal power series for ''f'' that uniquely determine the coefficients. The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
coefficients of the ''A''
''i'''s and ''b'' are
majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the ''A''
''i'''s and ''b'' has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge
where the scalar solution converges.
Higher-order Cauchy–Kovalevskaya theorem
If ''F'' and ''f''
''j'' are analytic functions near 0, then the
non-linear Cauchy problem
: