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== | In [[mathematics]], the '''Cauchy–Kowalevski theorem''' is the main local [[existence theorem|existence]] and uniqueness theorem for [[analytic function|analytic]] [[partial differential equation]]s associated with [[Cauchy initial value problem]]s. A special case was proved by {{harvs|authorlink=Augustin Cauchy|txt=yes|last=Cauchy|first=Augustin|year=1842}}, and the full result by {{harvs|txt=yes|authorlink=Sofia Kovalevskaya|first=Sophie |last=Kowalevski|year=1875}}. | ||
==First order Cauchy–Kowalevski 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]]s. The theorem and its proof are valid for analytic functions of either real or complex variables. | |||
== | Let ''K'' denote either the [[Field (mathematics)|fields]] of real or complex numbers, and let ''V=K''<sup>''m''</sup> and ''W=K''<sup>''n''</sup>. Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''−1</sub> be [[analytic function]]s defined on some [[Neighbourhood (mathematics)|neighbourhood]] of (0, 0) in ''V'' × ''W'' 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 [[Differential equations#Types of differential equations|quasilinear]] [[Cauchy problem]] | ||
:<math> \partial_{x_n}f = A_1(x,f) \partial_{x_1} f + \cdots + A_{n-1}(x,f)\partial_{x_{n-1}}f + b(x,f)\, </math> | |||
with initial condition | |||
= | :<math> f(x) = 0 </math> | ||
on the hypersurface | |||
= | :<math> x_n = 0 </math> | ||
has a unique analytic solution ''ƒ'' : ''W'' → ''V'' near 0. | |||
[[Lewy's example]] shows that the theorem is not 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''<sub>1</sub>, ..., ''A''<sub>''n''−1</sub> be [[analytic function]]s with values in [[endomorphism|End (''V'')]] and ''b'' an analytic function with values in ''V'', defined on some [[Neighbourhood (mathematics)|neighbourhood]] of (0, 0) in ''V'' × ''W''. 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]] coefficients of the ''A''<sub>''i''</sub>'s and ''b'' are [[majorization|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''<sub>''i''</sub>'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–Kowalevski theorem== | |||
If ''F'' and ''f''<sub>''j''</sub> are analytic functions near 0, then the [[non-linear]] Cauchy problem | |||
= | :<math> \partial_t^k h = F\left(x,t,\partial_t^j\,\partial_x^\alpha h \right),\text{ where }j<k\text{ and }|\alpha|+j\le k,\,</math> | ||
with initial conditions | |||
:<math> \partial_t^j h(x,0) = f_j(x),\qquad 0\le j<k,</math> | |||
has a unique analytic solution near 0. | |||
This follows from the first order problem by considering the derivatives of ''h'' appearing on the right hand side as components of a vector-valued function. | |||
===Example=== | |||
The [[heat equation]] | |||
:<math> \partial_t h = \partial_x^2 h \, </math> | |||
with the condition | |||
:<math>h(0,x) = {1\over 1+x^2}\text{ for }t = 0 \, </math> | |||
has a unique formal power series solution (expanded around (0, 0)). However this formal power series does not converge for any non-zero values of ''t'', so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |''α''| + ''j'' ≤ ''k'' above cannot be dropped. (This example is due to Kowalevski.) | |||
==The Cauchy–Kowalevski-Kashiwara theorem== | |||
There is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients, the [[Cauchy–Kowalevski-Kashiwara theorem]], due to | |||
{{harvs|txt=yes|authorlink=Masaki Kashiwara|first=Masaki |last=Kashiwara|year=1983}}. This theorem involves a [[cohomological]] formulation, presented in the language of [[D-modules]]. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a [[derived functor]] <math>Ext^1</math>. | |||
===Example=== | |||
Let <math>n\le m</math>. Set <math>Y=\{ x_1=\cdots=x_n \}</math>. The system <math>\partial_{x_i} f=g_i, i=1,...,n,</math> has a solution <math>f\in \mathbb C \{ x_1,...,x_m\}</math> if and only if the compatibility conditions <math>\partial_{x_i}g_j=\partial_{x_j}g_i</math> are verified. In order to have a unique solution we must include an initial condition <math>f|_Y=h</math>, where <math>h\in \mathbb C \{ x_{n+1},...,x_m\}</math>. | |||
==References== | |||
*{{citation|last=Cauchy|first=Augustin|year=1842|journal=Comptes rendus|volume=15}} Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58. | |||
*{{citation | last= Folland|first= Gerald B. | title= Introduction to Partial Differential Equations| year=1995| publisher= Princeton University Press| isbn = 0-691-04361-2}} | |||
*{{citation|mr=0717035|first=L.|last= Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer |year=1983|ISBN=3-540-12104-8 }} (linear case) | |||
*{{citation|first=M.|last= Kashiwara|title=Systems of microdifferential equations|series= Progress in Mathematics |volume= 34 |publisher= Birkhäuser |year=1983|ISBN=0817631380 }} | |||
*{{citation|last=von Kowalevsky|first= Sophie |journal = Journal für die reine und angewandte Mathematik|volume=80|year=1875|pages=1–32 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0080&DMDID=DMDLOG_0005|title=Zur Theorie der partiellen Differentialgleichung}} (German spelling of her surname used at that time.) | |||
*{{springer|id=C/c020920|title=Cauchy–Kovalevskaya theorem|first=A.M.|last= Nakhushev}} | |||
==External links== | |||
*[http://planetmath.org/encyclopedia/CauchyKowalewskiTheorem.html PlanetMath] | |||
{{DEFAULTSORT:Cauchy-Kowalevski theorem}} | |||
[[Category:Partial differential equations]] | |||
[[Category:Theorems in analysis]] | |||
Revision as of 06:03, 10 December 2013
In mathematics, the Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proved by Template:Harvs, and the full result by Template:Harvs.
First order Cauchy–Kowalevski theorem
This theorem is about the existence of solutions to a system of m differential equations in n dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables.
Let K denote either the fields of real or complex numbers, and let V=Km and W=Kn. Let A1, ..., An−1 be analytic functions defined on some neighbourhood of (0, 0) in V × W 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 Cauchy problem
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 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 A1, ..., An−1 be analytic functions with values in End (V) and b an analytic function with values in V, defined on some neighbourhood of (0, 0) in V × W. 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 coefficients of the Ai'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 Ai'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–Kowalevski theorem
If F and fj are analytic functions near 0, then the non-linear Cauchy problem
with initial conditions
has a unique analytic solution near 0.
This follows from the first order problem by considering the derivatives of h appearing on the right hand side as components of a vector-valued function.
Example
The heat equation
with the condition
has a unique formal power series solution (expanded around (0, 0)). However this formal power series does not converge for any non-zero values of t, so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |α| + j ≤ k above cannot be dropped. (This example is due to Kowalevski.)
The Cauchy–Kowalevski-Kashiwara theorem
There is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kowalevski-Kashiwara theorem, due to Template:Harvs. This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor .
Example
Let . Set . The system has a solution if and only if the compatibility conditions are verified. In order to have a unique solution we must include an initial condition , where .
References
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Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (linear case) - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (German spelling of her surname used at that time.) - Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
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